Cats are notoriously aloof, and that includes their typical reactions to being touched. Generally, cats are sensitive about who touches them and when -- particularly on their bellies. They are significantly more likely than dogs to reject your attempts to rub their tummies. That's because cats and dogs show their bellies for fundamentally different reasons. While it doesn't mean all cats dislike belly rubs, they don't instinctively appreciate them the same way dogs appear to.
Defending the Belly
The most sensitive part of the cat's body is his belly, as the soft flesh there protects most of his vital organs. This means that when he lays with his belly exposed, he is exposing his most vulnerable spot, and his instinct is to attack anything that touches it. Even if your cat is generally affectionate and loves when you pet him, he may instinctively swat and/or bite you if you try to give him a tummy rub.
When your cat lays around the house with his belly exposed, it isn't an invitation for you to reach out and give him a vigorous rub. Rather, it is a sign that he trusts you and feels safe in his environment -- he wouldn't display his stomach this way if he didn't feel secure. Attempting to pet his stomach may violate the trust that he has placed in you and his home and put him ill at ease. If your cat is showing off his belly, take is as a compliment but don't try anything.
Cats sometimes roll around on their backs and sides as a way of getting your attention, though not necessarily in the form of a belly rub. Affectionate cats love to capture the attention of their owners, and flopping around on their backs is one way that they typically do it. Petting your cat behind the ears and along the back reward him for his affection, but trying to rub his tummy may change his mood from loving to defensive.
Dogs Vs. Cats
Dogs and cats display their stomachs for different reasons. While dogs may do it as a sign of submission, or even because they just like the sensation of a tummy rub, a cat may do the opposite -- in the wild, when he shows his stomach, he is taking a defensive position. Cats in the wild are preyed upon by other animals, and they rely on their four sets of claws and their teeth to fend off would-be predators. When they lay on their backs or sides, it exposes their weak stomachs, but allows them to more ferociously defend themselves using all four paws and their mouths.
Testing the Waters
If you want to find out whether or not your cat enjoys a gentle tummy rub, proceed with caution. When he is laying on his side with his belly exposed, gingerly pet one of his front paws. If he doesn't react, do the same with a back paw. If he allows you to touch his paws like this, he may allow you to softly stroke his belly fur, as well. Be warned, though: Cats only allow the occasional belly rub if they are exceptionally comfortable with their owners. Don't take it personally if he greets your attempt with a swat -- he's only acting on instinct. And don't be surprised if your cat allows you to pet his belly for a moment, then inexplicably thrash and scat.