We have a similar arrangement. We went through all the household chores and figured out “who is good at this and doesn’t mind doing it?” So, for example, he washes the laundry, and I fold it. We each put our respective clothes away in the way we feel appropriate. And if that’s a pile on top of the dresser, that’s fine! Similarly, I will cook (I don’t like it, but I’m much better at following directions) provided he does the dishes and keeps the countertop clear.
When one of us isn’t able -- back issues kept me from bending, a kitchen remodel kept our dishwasher offline for months -- the other fills in, knowing it’s just temporary.
It does seem a more rational way of doing things. Why should the person who is ill suited for something do it just because of gender?