For an orientable $n$-manifold $M$ and its (orthonormal) frame bundle classifying map $\tau_M : M \to BO(n)$, we have a lift diagram of the following sort: ![Lift diagram here: the map from M to BO(n) has an up-to-homotopy lift through the projection BSO(n) to BO(n) ][1] There are two orientations on $M$. Is it ever true that they correspond to two different witnesses $H$ in the above diagram? This question is motivated by the observation that an orientation on $M$ is sometimes considered an example of an "$SO(n)$-structure" on $M$, a special case of a $G$-structure on $M$, part of the data of which one takes the 2-morphism witnessing the lift of $\tau_M$ to $BG$. [1]: https://i.sstatic.net/6Hx96CrB.png