Game Of Thrones: Is Tyrion Lannister Actually A Targaryen?
The Mad King
Tywin was Hand to King Aerys II Targaryen, otherwise known as The Mad King.
There are a couple of confirmed points regarding Aerys relationship with Tywin's wife, Joanna, that suggest he had some sort of relationship with her. Barristan Selmy tells Daenerys that the King was said to be infatuated with her, and that he took certain liberties at the bedding of her and Tywin.
While it's not revealed what those liberties were, it wouldn't be beyond the realms of Westeros for Aerys to have had sex (by consent or otherwise) with Joanna, and while that would've been too early for Tyrion, such activities could have continued beyond that point. After all, Aerys was King, and they could generally do as they pleased.
If Tywin knew or suspected this, then that would give him further motive to betray Aerys and help Robert Baratheon claim the throne instead, especially as he went so far as to have Prince Rhaegar's infant children murdered as well.