Theory twist:
So there's been a theory that Tyrion is actually a Targareyen, the son of the Mad King and Tywin's wife, which is why Tywin's hates him and says "you're no son of mine."
There have apparently been more hints in the book that the Mad King raped Tywin's wife at some point, but the hints sound more like before Tyrion would have been born since he's the youngest.
So what if Jamie and Cersei are the Mad King's kids rather than Tyrion. This would explain why Tywin hates Tyrion so much because the Mad King produced a great soldier and devious, intelligent (future) queen, while he produced a drunken dwarf that killed his wife in childbirth.
It would also fit with "whenever a Targaryen is born, the gods flip a coin" on whether they'll eventually be crazy or not. Jamie is sane, Cersei shows signs of maybe losing her sanity. 50/50
It would also explain why Tywin allowed Jamie to take the oath of the kings guard and give up his claim to the head of the Lannisters, because he isn't really a Lannister.
And the incest stuff. Mostly the invest stuff.