Gender and Party
There are many useful, oversimplified stereotypes that describe, in yin/yang fashion, the two main American political parties.  Here are some of them:

Democrats are feminine.  Republicans are masculine.
Democrats are nurturers.  Republicans are protectors.
Democrats are generous.  Republicans are stingy.
Democrats need a victim.  Republicans need an enemy.
Democrats are whiny.  Republicans are mean.
Democrats  are subjective.  Republicans are objective.
Democrats love equity.  Republicans love merit.
Democrats are formless.  Republicans are formal.
Democrats are fickle.  Republicans are loyal.
Democrats are revolutionary.  Republicans are evolutionary.
Democrats advocate for the powerless.  Republicans advocate for the powerful.

Opposite doesn’t always mean mutually exclusive.  Opposite can often mean complementary, where the two sides balance each other out.  In a given situation or transaction, one side may dominate.  But in a healthy, self-correcting society, over time, as the pendulum swings back and forth, there will be some kind of harmonization of the two sides, however fitful and episodic.

Democrats tend to be trusting, to assume the best about the intentions of others.  Republicans tend to be skeptical and on their guard.  Democrats tend to be forgiving with boundaries, and lax in their enforcement of the law.  Republicans tend to be firm with boundaries, and strict in their enforcement of the law.

Patriarchy and Feminism
Some people who call themselves feminist subscribe to the notion that patriarchy is an ill.  For many such people, categorical opposition to patriarchy is an essential component of their ideology, of their faith.  They won’t be happy unless/until patriarchy is abolished.  So they say.  So they believe.

But herein lies a great irony:  it takes a privileged person, one who is surrounded and protected by the fruits of patriarchy, to contemplate such abstract notions as the abolition of patriarchy.  Only the fortunate can afford to entertain such abstractions.

If you are ideologically opposed to patriarchy, if you believe it is a sickness or an aberration that our society, or our species, needs to transcend, and you are affiliated with a major political party, you are likely a Democrat.

But if you are comfortable with patriarchy, if you see it as natural, and you are affiliated with a party, chances are you are a Republican.