Well first I learnt that making a cylinder is relatively easier but making a toroid is way too difficult, the paper doesn't want to change its surface's curvature and IIRC there is a theorem like that by Gauss (??).Try it with a sheet of paper. Draw a line across the width on one side and along the length on the other.
You can turn it into a cylinder by bringing together two opposite edges and gluing them together, then into a torus by doing the same with the opposite ends of the cylinder.
You can do it either so that the first fold leaves the line inside as a circle and the outside line straight, or so that the outside line is a circle around the cylinder and the inside line is straight.
Check linkage of the lines in the finished torus in each case.
Anyway, to the conclusion, the blue circle are red circle are not linked like a chain in one but they are linked in folding the other way around as I expected. 100% their linkages are different, they have to be different!