Wow, that looks like a lot of work. Let's see what I can do on the figure.
Key ideas:
Translation, we just add the indicated values to x and y
Rotation CCW 90 (x,y) to (-y,x)
Rotation CW 90 (x,y) to (y,-x)
Reflection in x axis: (x,y) to (x,-y)
Reflection in y axis: (x,y) to (-x,y)
Rotation by 180: (x,y) to (-x,-y)
There was one that I didn't think had any path (and others if you look at the boxes that you don't get to) but I plowed through.
