My cat does the same thing (he is also neutered) I thought maybe he was just weird or it was really a sign of affection, but that would be intresting if your cat does have schizophrenia. I don't know if there is such a thing as a cat psycologist but you should reserch it, and if there is I would recommend it.
I am disabled and spend a lot of time sat in my lounge, my cat will lay on my knee for hours sometimes and be so content, and then she will get up and if I touch her or just move my hand she will hiss, spit and sometimes lash out with her paw (claws out sometimes) she doesn't always get down straight away either she will stand there and look at me with such evil eyes, then she will either lay down and go back to sleep or get down and go for a wander, and she will return later and be fine... it's so puzzling, could it be schizophrenia?
It could also actually be a skin condition....dry skin, etc. My cat does the same thing, and has rolling skin disease (which would have far more symptoms), but the point is he may have a skin condition that makes it difficult for him to tolerate petting or stroking for very long. He probably loves it, so seeks it out, but then it may hurt him. I would definately ask your vet about it. I now have my cat on a skin conditioner that I just put in his food, and it seems to help.
Good luck to you.
I've often thought it's the dichotomy of cat:
They love the attention, but have a threshold for too much. Sort of like pulling at a scab...it hurts like heck, but is so satisfying to pull off!!
Okay, bad analogy. :)
But you get the idea. We have one that you pet and eventually she'll sit there yowling and spitting...but still insisting you pet her.