How can I map a trapezoid onto another one with flipped axes. Assuming I know the coordinates of all the points (except $(x\prime,y\prime)$) how can I find $(x\prime,y\prime)$?
The drawing isn't to scale, its just to get my idea across.
If you extend $AB$ and $CD$ until they meet at a point $P$, and do the same for the primed vertices to get a point $P'$, are the angles at $P$ and $P'$ known to be the same? If so, then @gt6989b's solution will work OK. If not, then there's a theorem that says that for any two quadruples of points in the plane, there's a projective transformation taking one to the other.
And if you just want a MAP from one to the other, and don't care whether it's linear, projective, etc., then doing something like a bilinear map is probably your best bet. Finding a bilinear map or projective map between two quadrilaterals is pretty well described in multiple places on the web, so I'm not gonna repeat it here.
Create your transformation in the following steps:
Your map then is $f(x) = T(S(R(x)))$ and if you did it correctly, the 4 vertices will map out exactly. Then find your image with the same map.