Social organisation underground

Most of the underground societies in the Forges (dwarves, orcs, goblins) share a similar social structure: military and guild-like organisations are patriarchal, but property is matrilineal. Society as a whole is neither matriarchal nor patriarchal, but a mixture of both depending on context. It's a different take on the "men go hunting, women stay home and run the homestead" stereotype.
This plays out differently in different societies. The cliché in dwarven society, at least, is "women plan, men do," on the understanding that both of those things should work together; one isn't better than the other. How seriously people take this cliché probably varies with the individual; it's not like you couldn't find a canny male politician, or a short-tempered woman who likes to stab people.
This social dynamic works in a fairly non-dramatic way in orcish and dwarvish societies, which generally consist of small, interconnected communities, each of which has little hierarchy beyond councils of elders or similar. By contrast, the goblin kingdom of Five Bells has evolved a rather odd (to others) political structure. There is a war-leader who is always male, called the king, and a steward-leader who is always female, called the queen. Occasionally they are equal co-rulers, in which case they are symbolically married to each other. Often, though, only one of them is the "true" ruler (by heredity), in which case the other ruler is in a subordinate, advisory position. In this case they are called either the King and Land-queen (if the king is the true ruler), or the Queen and War-king. They might or might not be married to each other; they could well be married to other people.
This is a common way that civil wars begin in Five Bells; the non-ruling royal decides they'd prefer to be the one in charge.