I have a discrete set of points that describe a rectangular planar grid embedded in $R^3$.
I am also given one of the two normals to the plane and a graph specifying the connectivity of these points.
Assume that the winding order of the points in this grid is clockwise relative to the normal. I want to use a linear transformation to make the set of points have a counterclockwise winding order relative to the normal.
For example, if the grid spans the $(-1, -1)$ to $(1, 1)$ square centered at the origin and embedded in the $xy$ plane. Then for all points $p$ in the set one can do $p_x\rightarrow-p_x$ (i.e. mirror each point by the $x=0$ line). And this reflection will yield the opposite winding order.
I want to do this for any arbitrary plane, i.e. including those not containing the origin.
What rigid transformation do I use to achieve this?