Aaah I made this for fun just to see how my aliens stack up in the whole scheme of things against one another. I don't think this is all of them but I'm too tired to keep working on it so may I'll finish another time.
But this is basically a comparison of all the different species, who each have a distinct culture from each other, and how the genders get along in terms of equal opportunity to own property, have jobs, keep whatever income they make, how surnames are handled, how property is passed, et c.
Whatever sort of society and culture they have (matriarchal or patriarchal or egalitarian) stems from their pre-lingual/pre-sentient animal behavior.
For example Rélîqûn in their pre-lingual days traveled in pack-like groups lead mainly by the females, with one eldest matriarch more or less being the one who decided where to go and once speech came into play the matriarch would have final say in pretty much everything and was the unquestioned leader. Fast forward to when they have settled into an agrarian society, the females are still primarily the leaders of both government and military. Why change something that isn't broken? Males change their surnames to their wife's and legally she owns all his property even if he earned it.
Alternatively Ôrnês have always been pretty ornery and neither gender would take kindly to having one 'rule' over the other (physically they're nearly identical save those pesky sex organs). As such, both genders have the same rights in society and both can have any office in government of military they wish. I'm still working on how surnames will work for them, it's a bit more complicated when both wife and husband still own all their property and it stays separate even when married and after death (children are entitled to their deceased parent's property before the spouse).
I don't really need to explain how the patriarchies work, right? We all have firsthand experience of that. : D
But one interesting example of a patriarchy which isn't "males rule over females" is the Ŋóšt. Their species has low genetic diversity and male offspring are more commonly born than females. As such, the females are all sort of communally shared. Females reside only in special temples dedicated to the mother goddess and cannot be married to a single male. As such males marry each other, there is a 'third gender' of biological males who fit into a 'female' role in society. The females mate with any males that want children and if the child is male the fathers may take him home and raise him. However if the child is female she stays with her mother to be raised in the female temple, it is an immense honor to your family to sire a daughter even if the fathers rarely stay in contact or know her beyond name. Females own property, but are simply separate from the male society.
Eventually I'll write about all of them, but these are just some small examples~