Converting between coordinate systems

Hi all,

I mistakenly placed this in the .net forum originally.

My maths suck rather a lot, but I have a problem that I need to solve in code... it relates to the real world.

I have an X,Y coordinate system, in millimeters, that we may say is not rotated. The origin is in the center. Positive values are Up and Right.

I have another system, in millimeters, that we will say is rotated with respect to the first, but cannot be known before calibration.
This system has the same x-axis direction, but the Y axis is inverted (positive values down).

I can lay one over the other and calibrate them... (literally, in the real world)... and for any number of points necessary, I can know the coordinate location on each system.

Now I am looking for a methodology to take this calibration... and be able to know any point on the second system, when I know only the coordinates on the first.

Example: for point -123345, 150 : what coordinate is that on the second system?

Thank you all so much for your kind help.

R