Cats lie upside down for two very different reasons. A cat may lie upside down, often even hanging its head down when being held, to show complete trust and contentment in its surroundings. However, a cat may also lie belly-up as a sign that it's upset and ready to fight.
Since the belly is the most vulnerable part of a cat, many people mistakenly think that any time a cat lies upside down, it's showing its submissive side and wants to be petted. A cat in this position is happy and relaxed if she's also purring, if she appears content and if she sleeps in that position.
On the other hand, if a cat's other body language indicates anger, if her pupils are narrowed, her ears are flat or laid back or she is growling or hissing, the supine body position isn't a sign of relaxation, but a way the cat is preparing to defend itself against a threat. Paying close attention to a cat's total body language lets an owner know if the cat's feeling mellow or if it's better to leave her alone.