Every oriented vector bundle admits an orientation reversing isomorphism? Let $E$ be a real oriented vector bundle over a smooth manifold $M$. Is there a vector bundle isomorphism $\Phi:E \to E$ which reverses the orientation? (I know there are oriented manifolds with no orientation-reversing diffeomorphism, but this is not the same question, of course. Even when we specialize $E=TM$, I am allowing morphisms which are not differentials of maps ).

In general no, since such an isomorphism would reverse the sign of the Euler class. Consider $TS^2$ for a simple example where this yields two different isomorphism classes of vector bundles.