NecessaryScene points out that Neanderthals are extinct.
But I feel bad now, because I said "fortunately".
I meant meant "fortunately" in that at least it's one less civil rights issue to worry about.
But on the other hand, if they were still around, women (of both species) might be better off.
Women currently suffer from the problem that we don't otherwise seem to properly understand how to do civil rights for two groups that actually are fundamentally different in some way.
You see all these bad analogies to race, which misses the point that men and women are different in a fundamental way that people with different ancestry aren't.
If there was a clearly distinct second intelligent species, we'd have the necessary frameworks to handle them. And both groups including men, they should have been able to balance the rights. Including having different sports categories for the two species, if they had clearly differing physical abilities.
It would be a more even fight if it was "trans-species" Neanderthal men demanding to be let into the human male category (if humans were lower performance). Males versus males, not males versus females. I wasn't convinced patriarchy was a thing until I saw how readily women would be perceived as support humans in such a fight.
(Although it's possible neanderthal vs human might not be as clear-cut as sex is, if there was possible interbreeding. But it's an interesting thought experiment).