Why does a cat roll over to lie on its back when it sees you?

 

When you enter a room where a cat is lying asleep on the floor and you greet it with a few friendly words, it may respond by rolling over on its back, stretching out its legs as far as they will go, yawning, exercising its claws and gently twitching the tip of its tail. As it performs these actions, it stares at you, checking your mood. This is a cat's way of offering you a passively friendly reaction and it is something which is only done to close family intimates. Few cats would risk such a greeting if the person entering the room were a stranger, because the belly-up posture makes the animal highly vulnerable.
Indeed, this is the essence of its friendliness. The cat is saying, in effect, 'I roll over to show you my belly to demonstrate that I trust you enough to adopt this highly vulnerable posture in your presence."
A more active cat would rush over to you and start rubbing against you as a form of friendly greeting, but a cat in a lazy, sleepy mood prefers the belly-roll display. The yawning and stretching that accompany it reflect the sleepiness of the animal – a sleepiness which it is prepared to interrupt just so much and no more. The slight twitching of the tail indicates that there is a tiny element of conflict developing – a conflict between remaining stretched out and jumping up to approach the new arrival.
It is not always safe to assume that a cat making this belly-up display is prepared to allow you to stroke its soft underside. It may appear to be offering this option, but frequently an attempt to respond with a friendly hand is met with a swipe from an irritated paw. The belly region is so well protected by the cat that it finds contact there unpleasant, except in relationships where the cat and its human owner have developed a very high degree of social intimacy. Such a cat may trust its human family to do almost anything to it. But the more typical, wary cat draws the line at approaches to its softer parts.