See the original puzzle
I created a bunch of pictures in Mathematica to illustrate the process of turning the torus inside out.
If you look carefully at the pictures, you'll see that the rings remain linked throughout. And yet, at the end, the red ring is on the inside and the blue ring is on the outside. The resolution to this paradox is that the rings have switched places!
Update: Another intermediate step (hopefully this makes it a little clearer)