(This text was originally written as a comment, coming from a recent discussion with Harmonic System (@tslycha), both as further explanation of this perspective and as comparison, to our various ideas of gender and plurality.)
The description given in our account of gender is usually highly personal for us (in the relation between Seshedyt and Hypatia) and does not have the aspiration of general applicability. However, I do think I can say a number of things about this particular view on gender expression and identity, since it might be confusing in which way it is or is not related to the body, or to social functions; and want to begin that with the question of "social construction" or external influence, which seems to be so highly discussed today and is taken, at least by some people, as a given.
Social construction is an equivocal term, that is, it can mean many different things, and is often used to equivocate between them. To only give three examples: A skyscraper is socially constructed by engineers, an election by election workers, and our idea of justice by the justice system and the media; however, it is clear that the ways of "construction" of them are quite different. The skyscraper will be there, and clearly visible as something constructed, even if our society disappears; it is simply that, what was called before societal artifacts, and should be called that therefore to differentiate it from the others. An election is socially constructed in that it is a social process; the paper may be there afterwards, but the election isn't, it is purely processual in the society. And our idea of justice finally is not something in society at all, but within our minds, and only influenced by society. The problem here is, that we might easily confuse a social system with our idea of it, if we simply say something is "socially constructed"; but our idea of an election is clearly not the same as the election itself, and that is true also for more complicated cases.
When people say "gender is a social construct", it could mean three different things: a) gender is thing in the world we built; that is clearly wrong, since it either is something biological or sociological, but in neither case something built by a societal group, like a building is; there are not masterminds behind "gender" as a category, no masters of oppression or wise people or anything like that, all of that is clearly wrong if we look at the way that this category is older than the mechanisms necessary to control society on a large scale; b) gender is a social system: this makes more sense, and describes things like dating and beauty perception quite well; in this system, people take positions towards genders, these positions we might call "social genders". Social genders, as social categories, are assigned to the subject, that is the subjective system as a whole, with all of its headmates, since they all interact with society together; even in a more plural accepting society, a plural system would have externally one gender, that would be a compound gender expressing the differences of experiences between headmates (similar to genderfluidity, say), rather than having multiple genders at once (that is, it is a mixture, not a superposition). c) gender is a socially influenced idea: this also makes sense, but is now something that exists within the subjective system and is therefore alter-dependent. Every alter might be influenced to and be reacting to the social environment very differently, and need not have much connection to it to have this idea also.
Now there is a way in which people use the word "social construct" as an out out of conversations, such as saying "this is a social construct and therefore arbitrary and meaningless". The absurdity of this can not be overstated: by saying something is influenced by society, or built by it, they say it is meaningless. Are skyscrapers or elections meaningless? In some sense surely, but only in the sense that they are not thoughts. Are they arbitrary? Tell that to building regulators and election workers; it's absurd. They are not arbitrary. They are however, constructed by society, not by the Gods. Only if you consider everything, that is not divinely ordained, to be meaningless and arbitrary, you can actually believe in this. And I think the spread of this belief is quite astonishing as it is dangerous, as it also makes us blind to the very dangerious effect that meaning and purpose can have: that they need not be the meaning we approve of.
The one case left is that of societal influence on thoughts, and here we can see more clearly where this idea originates - from structuralist linguistics - but also, how it was wrongly expanded onto all of realty. At the same time, there are extra-lingual uses where this kind of view makes sense, and that is in the general view of social systems/signifiers, which is worth getting into, as one of the characteristics of our view of gender is the almost complete absense of these kinds of signifiers, which certainly needs to be explained.
But first of all, what does the theory of social construction of thoughts get right about them? Mainly, that concepts are historical and therefore changing depending on the historical material that is to analyze (even if they don't appear to change, broader applicability does change the extension of any term, and therefore in some way this idea itself, as it might loose associated qualities, assumed substantial); and that this history is not one that can be written outside of that history itself, that we are active participants, or co-constructors, of that future on which the further existence of our own ideas, or their change, depends. What it massively gets wrong is the assumption of "society" as a voluntarist super-subject that can arbitrarily "decide" on meanings by a general societal will; what is wrong about it, in other words, is its Russeauist attitude.
But that is not all of it. There is a certain sense in which social construction does describe something correctly, but precisely in the way in which it itself dismisses it. Because, there are truly two different versions of the hypothesis, or description, of socially constructed ideas: on the one hand, the idea that our very desires or motives are shaped externally, that we are in ourselves nothing etc, and on the other hand, the idea that there is this external influence that we experience as a violation of our own subjectivity, which does prexist in some form. The difference, in other words, is if we ourselves are considered to be thus constructed, or only our idea of who we ourselves are, in specific aspects that are external to our selves, and that we can understand and come to an understanding of what was behind/beneath it.
But no matter how we actually may think about this - to what extent our ideas are influenced by socialization, or to what extent our thoughts really are our own - what I really want to plea for is to please stop the use of "social construction" as an argument, since it really doesn't prove or mean anything. Only in an environment where ideas that are in some sense influenced by society would be surprising, it would even be noteworthy to state it. Therefore, the common use of social construction as a "gotcha" is not part of enlightenment, but of the anti-intellectual usage of dishonest surprise, of the overuse of affectionated shock for the sake of marketing etc, and has no real place in any serious theoretical discussion. If what is meant, is that a certain idea is historical, or influenced by its societal surroundings, than that can be said and should not be surprising to anyone; and those who are surprised, too often only act in this sense too in the other direction. I don't think the glee in some progressives eyes, when proclaiming that X is "merely a social construct" is in any way more honest than the resistance against it; and if by that, one person might want to suggest, that it's a bad construct, than it is more honest and sensible to simply say, that our systems of socialization and education are flawed, rather than to state, as if surprised by it, that they exist.
These are all very general statements; what does this to do specifically with gender and plurality? Well, gender exists, at least partially, as something that can be called a social construct, as we said, and specifically as either a societal system or a concept that has influence from socialization. Importantly, those elements are orthogonal to each other, that is, the social system can be understood both in itself as an objective system, as well as in its representation as idea, or its expression in language, and the thought can be seen to be influenced by its expression (that is, by its social element of communication and relation to other people) or not, and the same is true for the subjective idea of gender as existing for each alter, which can have a social influence or not, and can also be seen in relation to the social system of gender or not. In other words, whereas "social construction" as "influenced by socialization" is a mode of thought, that can be used to analyze any semantic or intentional content, "social construction" as in "the societal system" is merely a specific object of analysis, namely the systems of society, in sociology.
What does this now have to do with the idea of the arbitrariness of the symbol? Now here we get a bit closer to something that is true of this sentiment, but only very partially. It is indeed true that words, or social symbols, are in some sense arbitrary, but not in a very strong way. Their arbitrariness is only in the use of the symbol, as different to other symbols, to signify meaning. I will not go into the details of structuralist linguistic here, but suffice it to say, I think the main idea of it is correct, but only for lexicography, not for semantics, grammar or interpretation. And therefore, its application on other systems only works where elements are used with symbols of lexical quality, that is, that they represent something in as indirect a way as a sound or scribble represents the meaning of that word. And here, we indeed have something we can describe under the category of gender; and we can say that the use of gender either as phenomenon or symbol can describe much of the difficulty of dealing with it as a concept on the whole, and the way it is generally related to body and leib.
The symbolic notion of gender is what you could also call symbolic gender, or external gender identity. This can include a variety of things, from relations to sex characteristics (which need not be directly bivalent or correlated), various forms of expression like clothing or general appearance, the use of these expressions in behaviors, such as ways of talking and moving, the relation of that to gender roles in specific social relationships, and the agreements or disagreements to these roles as expressed in external identifications by the use of labels or social identity terms (which are vocalizations of these attitudes, including also the understanding of the other categories, which hence are, but only linguistically, subordinated to this use in caregorizations of evaluation). Because these are symbolic, the contents of them are, to some extent, arbitrary; in the same sense as the sounds in the word "house" or "tree" could be changed and are different in a different language, so can social roles or the connotations of clothings or of bodily postures; but their functions are not arbitrary, and they are also pretty much determined within a certain historical and cultural setting. Their arbitrariness is primarily one to the phenomenological understanding, not to the system itself. This is because all of this symbolic use of gender is opposed to, but also used by, a different, phenomenological understanding of gender.
In this understanding, the body, expressions, attitutes, roles and identities also appear, but now without the symbic externalization. The body appears not as leib, but as a relationship to the leib that is to-someone, but in a way I don't totally understand, since the other is not present outside the symbolic form; the expressions and attitudes of this gendered body are actions of a subjective desire, not of an objective goal of symbolization; and therefore, the corresponding roles and identities are not necessary for them, but on the other side, are hermeneutic practices to understand those phenomena. I don't set myself a goal to "socially be a woman" and act accordingly, but simply observe that my expressions happen to fall under that term, and accept this identity, as by an experimental hypothesis, minus the calculated use of means of measurement or falsification (which is hard to do for matters of bodily functions, only for the goal of finding something like that out; those experimentations ususally have more than epistemic desires connected to them). In these ways, I create a gender that creates, from the experience and projections of my experience of the leib, an idea of "body" that is not external; and I would now posit that it is precisely this "body" that is the foundational signifier in the social symbolic exchange of gender.
Because, what is the goal of gendered exchanges if not "getting to the authentic reality"? It is such a cliche, but there is truth about it, if you take it as more than a cliche. There is no authentic reality in either phenomenal life or external measurement, since both are so full of doxa and rarely see the light of an idea; but that gendered body, that is the element of subjective identification, is neither leib nor body; it is the body of the leib, if you want. It is that what is for the self of itself symbolic, but not linguistically expressible. And that precisely because it is, in a basic sense, invisible or empty.
However, since the other in this exchange has no direct understanding of the indirect nature of this "body", it is identified with the actual body, without any connection to the leib. This is what might be the basis of the strange effect of "objectification"; that we find it weird to treat an object, the body, like an object, as if it would not have been one.
This is now also related to plurality. I have not developed a very strong theory about it, but I can give a few broad strokes of the types of relations I am thinking about: Every headmate has their own interal understanding of their body, as in the inner world or at least like they externalize that understanding (drawings, voices etc.), but they only have one leib to experience this through; at the same time, this one leib is a connection of various different modes of experience, which's activation patterns (like which senses are strong, or how much overwhelmed etc.) might create actually a dissimilitude of various leibs, with a connection in that external body they are modes of. So internally, different bodies are connected to a leib, but externally, different leibs to a body. And each of them has their own connection to gender. One can see now that the leib here is closer connected to that internal understanding of gender, giving rise to the symbolic "body" of each respective leib, whereas the body is in itself already the basis of the identifying act of symbolic gendering, identifying body and "body". (For example, in identifying the symbolic use of gender identifies in the inner world (the purely symbolic gender of the internal body) with the gendered experience of that alter fronting (the "body" as the symbolic self-interpretation of that alter's leib-state), as participating in the external gendered structure of the system (as a property of the external body those leibs are connected to).)
This is yet a pretty broad strokes theory, but I do think it is interesting enough to think about it, how it could supplement the otherwise a bit too self-satisfied view of social constructivism, that external reprsentations and their occasional arbitrariness are enough to disregard subjective realities. And maybe, in this process of inner self-mediation of symbolic or conceptual and phenomenal or concrete thought, it could also shed some light to this most peculiar element of modern metaphysics, that my thinking in some way has always been circleing around: the linguistic denial of substantiality, or the absurdity of conceptual thought, to think of its own means of conception as less appropriate to reality than the result that has been reached by them.
Note: I want to add to this text that the whole text only is about certain elements of gender, not of all that can be considered to be gender. Specifically, it is about the subjective experience of gender identity as externalized and internalized to the consciousness of each alter, not about external qualities observed on the expression or characteristics of the body. These properties, while also subject to subjective conceptual categorizations that are ultimately influenced too by various social and symbolic identities, have different dynamics than the subjective self/ves-understandings themselves. This is notable in political contexts, where these external identifications dominate, and such projects, as I often naturally tend to, to understand the posed questions in a more subjective way, will often end up failing not for reasons of internal inconsistency, but by the nature of these external systems that are constitutively misaligned to represent subjective experiences in other categories than they could self-describe their own reality accurately.