Here it is in a nutshell:
I dunno; it sounds like you get them pretty good. 
This too. I have a cat, and she’s my little buddy, but she is totally an asshole.
Cats are the devil. Mind you, the devil is said to have to power to assume pleasing shapes.
My recently passed-away kitteh used to purr while fighting with me and the harder I smacked her upside the head, the happier she became. Now *that *is psychotic behaviour!
I’ve also known a few cats who get so stimulated by a good petting that they switch over from happy to fighty because an energized cat is one that wants to kill stuff. And is prety much their psyche in a nutshell: awake = looking to commit murder and eat the victim.
They’re so adorable. 
Looking at comments here, cat psychology is relatively straightforward. What I don’t get is why people would voluntarily keep cats?
Neither do the cats.
Yeah, the simple truth is that cats are nature’s little psychopaths.
That’s why when I see Scar rolling onto his back to show his belly, I reach for my leather work gloves. It makes the scritchy-bitey game so much more fun. Normally, he rolls over like that, gets some scritchens, then attacks and gets to walk away victorious when we pull back in fright. But when the glove goes on, he can’t hurt me…so he walks away confused.
I think the artist here has captured cat moods, and the last image is exactly what the OP is talking about. I’ve had cats like that. They’re not being mean and they’re not upset. That curl around grab is just their reaction when they’re petted. They think that you’re asking them to play that way.
What really sucks is when they start embezzling from you.
The time to worry is when you DO start getting cat psychology.
In Russia, cat psychology gets you.
I give the WEB (Wee Evil Beastie) treats when she asks me for some, I open the door when she wants in or out, and I apply scritches when she rubs against me. She hasn’t murdered me in my sleep so far. Some call this a stalemate. I consider it a win/win.
My two favorite explanations of cat behavior:
Dog: The humans shelter me, protect me and feed me! They must be gods!
Cat: The humans shelter me, protect me and feed me! I must be a god!
Or…
A cat does not want to murder you because of your gender, or color, or sexual orientation or nationality. A cat wants to murder you because he is a cat.
Obligatory First World Problems Cat link.
To your dog, you are the FOOD GOD.
To your cat, you are just FOOD.
In this context, you will find that nearly everything about canine and feline psychology makes almost perfect sense.
Stranger
I don’t understand the animosity here towards cats by supposed cat owners. ?? I’ve had many cats and also volunteer at a shelter. In that time I’ve run across very few cats who actually like their tummies rubbed. *So I don’t do it. * If you’re getting your hand attacked, your cat doesn’t like it and has humored you for as long as they can stand. So don’t do it.
Like one poster upthread, I’ve never gotten myself mangled or chewed up, not even by scared shelter kitties. Maybe because I appreciate and respect cats for being cats, are and don’t think of them as evil or monsters.
And Cat Whisperer, I’m really surprised at this:
She has her reasons for what she does, and they’re not to piss you off. She’s a cat and does what accomplishes what she wants. That it doesn’t please you has nothing to do with it. With your ID, I thought you’d understand this. She’s not an asshole, she’s a cat. Doing cat things.
But you were brushing the cats belly, right?
Cat bellies are a problem area. I have learned not to touch it. I think rubbing a cat’s belly is comparable to stroking the sides, or underfoot, of a human. It feels nice untill, suddenly, it tickles and you can’t help but shudder and spasm. In the case of the cat, the spasm takes the form on an, mostly involuntary, bite. Usually the cat herself is most upset that she suddenly started biting you, and her response is to jump away from your lap. It is her way of saying: “OMG, what just happened? What did I do ?” .
If she then starts to sit a few yards away from you, with her back turned to you, that is her way of asking forgiveness. She is not moping, that is a human interpretation. Being near you but not subjecting you to an intimidating stare is a cat’s way of saying: “Please, can we make up?” You’ll see that is what she intends if you make a kind “come here”-sound. Usually the cat sitting with her back to you will happily turn around and come back to your lap.
And while I am at it, here’s a quick compendium of cat speak:
Look your cat in the eye with a relaxed look, eyelids somewhat closed, then slowly blink at your cat. Close your eyes slowly for about a second, then slowly open them again. If your cat returns the gesture, usually followed by casting down her eyes, that is the cat equivalent of a friendly smile exchange. It means: I’m so relaxed around you, I even dare to sleepily close my eyes near you. Here’s a short YouTube clip showing the slow blink.
Rapidly swishing tail. cat is not quite happy and making up her mind to do something else or go somewhere else. In a few seconds, she’ll start protesting or she’ll go away.
If your cat wants to snuggle or is hogging the paper, there is a quick way of telling her you’re not in the mood. Blow at her. Not hissing, hissing is too strong. But blowing at her face is a good way of telling her: no, not now, go away.
Nice set of tips there Maastricht.
Much appreciated.
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Cats are alien creatures and should be treated as such. Canines co-evolved with us socially. Cats did not. If you die in your home alone with a dog, she will wail and bark and freak out. If you die alone with a cat, she will eventually eat you.
But, that having been said, as an alien on Planet Earth, I like cats better then dogs.
Here are some more YouTube tutorials on cat body language.