Let $p \in M$ be a point of a non-orientable smooth manifold, $M$. Does there exist a diffeomorphism $f: M \rightarrow M$ with $p \mapsto p$ and such that $df : T_pM \rightarrow T_pM$ is orientation reversing? My feeling is yes. I was thinking about trying to take an embedding $\gamma : S^1 \rightarrow M, \star \mapsto p$ such that parallel translation around $\gamma$ reverses orientation, then pushing forward the vector field $d/d\theta$ and extending it. Then taking the flow at time $2\pi$. However I wasn't sure about the existence of such a $\gamma$ and the whole approach seems a bit contrived. Is it true and if so is there an easier way? Thank you for your time.
P.S.this is motivated by the question of the well-definedness of connect-sum for non-orientable manifolds.