It is well known (using for instance sheaf cohomology) that there exist only two possible one-$\mathbb R$-dimensional vector bundles over $S^1$: the trivial bundle and the Möbius bundle. But what about the Möbius bands with not one, but $n\in \mathbb N$ half-twists? It is not obvious to me why these are not distinct line bundles. I am also not entirely comfortable with sheaf cohomology arguments.
This is also physically motivated; if you make a Möbius band, it's easy to see that two half-twists don't "cancel" to make a trivial bundle; so why do they "cancel" topologically?