Alter a map to be orientation preserving I'm reading the h-cobordism by Milnor and he claims that we can alter a map to make it orientation preserving(second paragraph in page 58). I'll give a detailed description in the following:
Suppose $f:\mathbb{R}^n\to \mathbb{R}^n$ an orientation reversing  diffeomorphism. And $\xi, \hat{\xi}$ two vector fields in $\mathbb{R}^n$($n\geq2$).
My question is: if $f_*\hat{\xi}=\xi$, is there exist an orientation preserving diffeomorphism $g$ from $\mathbb{R}^n$ to itself such that $g_*\hat{\xi}=\xi$?
I think if I can take a map that change the direction of a vector field orthogonal to the one above, this problem will be done. Nevertheless I'm not sure there a correspondent modification of map to the modification in tangent space.
Thanks in advance.
edit: In the situation about the Milnor's book, it's enough to consider only the gradient like vector field in a morse chart, more explicitly $(-x_1,…,-x_\lambda, x_{\lambda+1},…,x_n)$so reflection about first coordinate will be an answer. Maybe the answer to this question is negative
in general case.
 A: If there are no restrictions on the vectorfields then there is a counter example in dimension 2. Let $f(x,y)=(x,-y)$, $\xi_1(x,y)=(-y,x)$ ( or if you prefer $\xi_1(x,y)=-y\partial_{x}+x\partial_{y}$ ) and $\xi_2(x,y)=(y,-x)$. Then $f_*(\xi_1)=\xi_2$, and $f$ is orientation reversing. Now suppose that we have an orientation preserving diffeomorphism $g$ such that $g_*(\xi_1)=\xi_2$. Notice, that $g$ has to map the origin to the origin. Since $g_*$ maps $\xi_1$ to $\xi_2$ we can check that if $\gamma(t)$ is an integral curve of $\xi_1$, then $g(\gamma(t))$ is an integral curve of $\xi_2$. Now the integral curves of $\xi_1$ are just circles around the origin going counterclockwise and the integral curves of $\xi_2$ are circles around the origin going clockwise. Now let $\gamma(t)$ be an integral curve of $\xi_1$, then $g(\gamma)$ is an integral curve of $\xi_2$, thus it has winding number $-1$ around the origin, but $\gamma$ has winding number $1$ around the origin, and $g$ is orientation preserving so the winding number of $g(\gamma)$ has to be non-negative, and we have a contradiction.
