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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 ).

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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.

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