It gets weirder: https://www.youtube.com/watch?v=8rqBFvm2lME

And even weirder: https://www.youtube.com/watch?v=LLw3BaliDUQ

True, that recommendation to look at the video.

I found the example in Feynman of using a single "ribbon" (a belt) connected to a book to be a much better example of a 720° rotational symmetry than this very cool and mesmerizing but also brain breaking video clip of the "ribbon cube"!

There's no trick. It's really unintuitive but it works.

Seems like the top and bottom ribbons would work. But the ribbons on the side get a kink every 2 rotations that the animation sort of sweeps under the rug?

They get a twist, but then loop around the cube, which undoes the twist. You can see their full orientation throughout the animation. (Or try getting your own cube and ribbons and try physically turning it to see for yourself.)

You can see the key behavior by fixing one end of a belt, and moving the free end.

You should find that there is a way to apply and remove double twists in the belt without twisting the free end.