Abstractly, on the topological circle $S^1$ there are only two real line bundles, up to isomorphism: the trivial one $\mathcal{O}$ and the Moebius strip $\mathcal{O}(1)$ (thinking of $S^1$ as $\mathbb{RP}^1$). So we have $\mathcal{O}(2k)\cong\mathcal{O}$ and $\mathcal{O}(2k+1)\cong\mathcal{O}(1)$ for any $k$.