If you name the verticies of the four rectangles A, B, C, and D.
The rectangle would appear thus:
The rectangle would have four transformations:
Rotation of 0º (or 360º) : $ R_{180} $
Rotation of 180º: $ R_{0} $
Flip About a Horizontal Axis: H
Flip About a Vertical Axis: V
From this, the Cayley table can easily be constructed.
