Several key points:
1) There are two distinct types, we call masculine and feminine; each actual man or woman will have some configuration of both types, but men tend to have more masculine than feminine, and women tend to have more feminine than masculine, although that isnt always the case.
2) Those types are also generalizations. At bottom there are no genders, gender does not go "all the way down" (there is the fact that we all start out the same gender at the biological level, but more significantly consciousness itself, pure self-aware sentient subjectivity, is non-gendered; gender divisions, which do exist, impose themselves upon consciousness but do not grow up from within it).
3) Gender is not merely a contingent social construction, because gender differences reduce to biological differences; while in embryo they start out the same, man and women have different chromosomes and also tend to have somewhat different paths for processing information in the brain. While it is true that gender is part social construction and that this construction is indeed somewhat contingent and should be improved over time (made less arbitrary, less oppressive and irrational), gender differences are also not just the result of mere social contingency of forces but are rooted in biology.
4) There are deep, philosophical level confusions about gender because of these complexities as just noted; gender is both innate and socially constructed, men and women are both genderized to their general gender norms but also are individual and cannot be totally generalized like that; masculine and feminine, as psychological types arising from biological distinctions and socially constructed forces over time, do not perfectly map onto "male and female"; and even though gender is a deep and irremovable part of who we are as individuals, consciousness itself at the metaphysical or ontological level is non-gendered.
. . .
Those differences in (4) make the situation confusion, and make a philosophy of gender difficult. How exactly to tease apart one thing from another? Where specifically does social construction of gender identities end and innate biology begin? Likely there is no hard line there, but a porous kind of membrane through which things pass in either direction, each influencing or limiting the other, and the vagueness of this situation belies our ability to really grasp it and determine clear entities and causes here.
We share a lot of our "genetic memory" and "cultural memory" with those of the opposite gender, but it must also be the case that some genetic and cultural memories are gender-specific; namely, women must have some genetic and cultural memories (residual influences at the instinctive, memetic or sociological levels) that men do not have and vice versa. Again this is another difficult confusion since there is no clear way to draw a line separating the one from the other.
Since we cannot yet form a philosophy of gender, we can at least make statements about what we know for certain (philosophically speaking) gender is not:
1) Gender is not absolute (there is no "one male gender" and "one female gender", each gender is more like a general collection of various traits and tendencies, various shades of differences).
2) Gender is neither entirely biologically innate nor entirely socially arbitrarily constructed, but a combination of both of these.
3) Gender is not subjectively universalized (I mean that gender does not go "all the way down" in determining subjectivity-consciousness; gender is something that is imposed upon this consciousness and is somewhat formative of subjectivity, but does not arise absolutely from within these; there is a sense in which male and female subjectivity-consciousness are identical to each other (at the metaphysical or ontological level, for instance)).