basking in the sun may lie on his back at times, but in interactions with humans or other animals, “belly up” is bad news. It means the cat is the loser in a two-cat brawl.
It frustrates many cat owners that cats, quite unlike dogs, don’t usually like to be held on their backs while their bellies are rubbed. But, obviously, the cat associates “on my back” with “I’m in the middle of a ﬁght!” It is not a position with pleasant associations. If you’re fortunate enough to have a cat who willingly exposes his belly for you to rub, you can assume the cat trusts you totally.