It is well-known that $SO(2)$-principal bundles over a manifold $M$ are topologically characterized by their first Chern class. I was wondering what was the characterization of $O(2)$-bundles in terms of characteristic classes. I guess the first and second Setiefel-Whitney classes are necessary for the topological characterization of $O(2)$-bundles, but they can't be enough, because if $w_{1} = 0$ then one should recover the classification of $SO(2)$-bundles, which is given by the first Chern class and not by the second Stiefel-Whitney class.
Thanks.
We have a fiber sequeunce $BSO(2)\to BO(2)\to B\mathbb{Z}/(2)$, and so we have that a $O(2)$-bundle, or a map $X\to BO(2)$ factors $X\to BSO(2)$ if and only if the composite $X\to B\mathbb{Z}/(2)$ is null-homotopic. $Hom_{\mathcal{h}Top}(X, B\mathbb{Z}/(2))=H^1(X; \mathbb{Z}/(2))$, so we have a class $x\in H^1(X; \mathbb{Z}/(2))$ representing the isomorphism class of the bundle. This class is clearly the pullback of the the universal class $x\in H^1(BO(2); \mathbb{Z}/(2))=(\mathbb{Z}/(2)[w_1, w_2])_{deg=1}=\mathbb{Z}/(2)w_1$. We note that since $O(2)\neq SO(2)\times \mathbb{Z}/(2)$, this class cannot vanish identically, so that $x=w_1$. Now we are left with an $SO(2)$-bundle, which you already know about!