I asked this question on Math.SE as I thought the perspective of representation theory might be enlightening.
But since the question was provoked by a description of Spinors describing the spin of electrons in Dr Tongs notes where he described that 'one has to walk around an electron twice' for it to return to the same position; I thought I'd also ask it here.
Consider a Möbius strip; draw on one side of it an arrow aligned vertically; now take it for a trip by around the strip; then when it comes back to the same position it has flipped direction; another circumnavigation of the strip returns it to the right way up.
Now Spinors have to be rotated twice to return it to the same position.
Can these two pictures be connected in some way?
There is also this plate and belt trick; which might or might not be connected.