On the real $\textbf P^1$ we have these algebraic line bundles: $\mathscr O(1)$ and $\mathscr O(-1)$.
Which one corresponds to the Möbius strip? (Both are $1$-twists of $\textbf P^1\times\textbf A^1$, so how to distinguish them? Yes, by means of their transition functions, but how do they tell me which one has the Möbius strip as total space?) And what is the total space of the other one?
I can only imagine that non-orientability should correspond to the absence of global sections, so I would bet on $-1$, but with no real reason.
Also, I can't figure if to every $\mathscr O(d)$ there corresponds a different total space, or there are repetitions. Of course, by viewing those bundles as holomorphic bundles, there are only two surfaces up to diffeomorphism. But what about the algebraic category?
Thank you for any help!