I apologize for this post, because I may've just figured out something that's been obvious to everyone else.
I've been casually thinking for years about Engel's observation, "The first class opposition that appears in history coincides with the development of the antagonism between man and woman in monogamous marriage, and the first class oppression coincides with that of the female sex by the male."
I kept thinking there was no connection other than a love of a hierarchy.
But that assumes loving hierarchies is natural.
This morning I saw what I had missed. What underlies sexism and classism? Cheap labor, in the form of exploitation based on sex and class.
If you're exploiting people, I only see two basic choices:
1. You think it is bad to exploit people, and you stop.
2. You think it is good to exploit people, and you continue.
The easiest way to think it is good to exploit is to adopt a belief system that assumes exploitation comes from a natural hierarchy. Sexism, classism, xenophobia, racism, and religious bigotry are all rationalizations for paying a group of people less, treating them worse, and taking whatever you can from them.
ETA: This post partly came from thinking about the tragedy of the Europeans coming to the Americas: the Europeans met a representative sample of Americans, but the Europeans were not representative. The leaders came to exploit.