My duck likes to be held on its back?!
I've only had ducks for a little under two months, but I've had chickens before like that. chickens and some other birds (some individuals more than others) mellow out on their back. some don't especially like having that done to them (since it makes them feel a bit vulnerable I suspect), but some seem to like getting flipped on their back and held like that, at least when they're being handled by someone they trust. and pretty much each critter can have their own quirks in how they interact with people and/or other critters, some rather odd.