I have read multiple cat care books as an old dog guy and a new cat owner over the summer. Your old large dog might spin in circles before it goes to sleep (thought to be instinctive about building a nest). My dogs always flopped, sighed, farted and then went off to dreamland without a care in the world. Not so with cats - they have different levels of rest and sleep depending on how much of their brains they want to shut down.
Cats will often catch some rest on their haunches with not a lot of their brains shut down. They will do this on their haunches with their eyes closed and their head up - like the Sphinx in Egypt. They are partially asleep but not at complete rest. They are able to be instantly awake to defend themselves if danger presents itself. They will ignore familiar non-threatening noises but a "touch" will have them instantly alert.
Deep sleep when a cat is curled up or laid out is different. Most of the brain has been shut down so that the cat can catch meaningful rest. Nevertheless, the sleeping cat will have it's ears perked with a small part of it's brain alert to sound. The cat won't care if the TV is playing. In bed with you at 3 a.m. the cat won't care if an airplane flies over the house or if you fart. Those are familiar noises and nothing to worry about. If however a large dog starts barking on your front lawn at 3 a.m. the cat will immediately wake up and try to make a decision about fighting or fleeing. That cat will be wired for action one way or another in an instant.
Anyway I caught Mister Mittens doing the "Sphinx Thing" this afternoon while I was taking a beating at the Big A: