How do I negate the effect that the rotation of a plane has on a rectangle?

I would define the rectangle by the cooridnates of its lower left (x1, y1) and upper right corner (x2,y2). Any rectangle can be represented this way. I specity the rotation by angle $\theta$.

When a point (x,y) gets rotated by angle $\theta$, its new coordinates are:
\[x\cos(\theta) - y\sin(\theta)      \text{and}    y\cos(\theta) + x\sin(\theta)\]

To put the rectangle back, you need to rotate its coordinates by angle $-\theta$.  Thus, to put the aforementioned rectangle back where it was before rotation the lower left corner will be at

x-coordinate: $x_1\cos(-\theta) - y_1\sin(-\theta) = x_1\cos(\theta) + y_1\sin(\theta)$
y-coordinate: $ y\cos(-\theta) + x\sin(-\theta) = y\cos(\theta) - x\sin(\theta)$

and similarly you can compute the coordinates for the upper right corner.