Can I flip orientation at a point of a non-orientable manifold? 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.
 A: Let $M$ be a non-orientable smooth manifold. Non-orientability means that there are two open sets $U,V\subset M$ diffeomorphic to $\mathbb R^n$, orientations $\zeta$ on $U$, $\xi$ on $V$ and two points $p,q\in U\cap V$ such that $\zeta_p\ne\xi_p$ and $\zeta_q=\xi_q$.
Now pick a compact connected nghd $K\subset U$ of the pair $p,q$ and a diffeotopy $f_t:M\to M$ that is the identity off $K$ and moves $p$ to $q$: $f_1(p)=q$. Since $f_t|U$ is a diffeotopy of $U$ and  diffeotopies preserve orientations, we have $f_{1,*}(\zeta_p)=\zeta_q$. Similarly, pick a compact connected nghd $L\subset V$ of $p,q$ and a difeotopy $g_t:M\to M$, the identity off $L$ with $g_1(q)=p$, to have $g_{1,*}(\xi_q)=\xi_p$. Then the diffeo
$h=g_1\circ f_1$ does the job:
$$
h_*(\zeta_p)=g_{1,*}(f_{1,*}(\zeta_p))=g_{1,*}(\zeta_q)=g_{1,*}(\xi_q)=\xi_p\ne\zeta_p.
$$
For a flip at any other point $x\in M$, take a diffeo $\varphi:M\to M$ with $\varphi(x)=p$ and $\ \varphi^{-1}\!\circ h\circ\varphi$ reverses orientation at $x$.
