If the guy finds it down, he puts it up. If the woman finds it up, she puts it down. Putting it up or down is the same amount of work, so I see no rationale in a couple for forcing it to be put either up or down.

Furthermore, it's the same amount of work to move the seat before or after it's used, it's just a question of who's doing it, so if you are moving the toilet seat after you're done it's a partial waste of time because the next person to use it might want it the same way you had it.

If a woman insists you put the toilet seat down, this implies she thinks her time is more valuable than yours, so unless it really is, I would take this as a very bad sign.