Ah, the Time-Twister.
In the original series, there is a device called a “Time-Turner” that can magically transport you back in time by one hour. That has huge implications about the nature of the universe, and the books never specify its limitations. Logically, you'd expect them to be a huge issue – the Aurors would all have them, Voldemort would probably have one, and every important battle would be based on tricky uses of time travel.
I had three choices: I could justify Time-Turners, I could eliminate them entirely, or I could change them into something else. Harry Potter and the Methods of Rationality takes the first option; it creates a (supposedly) consistent set of rules for how Time-Turners work and how they handle temporal paradoxes. I didn't want to do that, because this isn't a story about time shenanigans. It's more of a story about characters and their emotional interactions. So it was between eliminating Time-Turners and changing them.
I didn't want to eliminate them because I really really liked the symbolism of Hermione using the Time-Turner to take extra classes. Ze warped the very fabric of time in order to study harder – that's some serious dedication. So I kept the object, but changed it. The Time-Twister doesn't send you back in time by an hour... it just gives you an extra hour, somehow, and no one's exactly sure how it works. You can use it go to two classes that are at the same time, but if you do, you won't remember which order you went to them in, and if someone follows you around, they'll come to both classes with you – so the Time-Twister doesn't just affect you, it affects everyone around you.
(Another reason I made Time-Twisters work this way is that it means I don't have to worry too much about keeping a consistent timeline. If things seem inconsistent or anachronistic, they are! It's canon!)