Let denote reflection about the y-axis and denote counter clockwise rotation by the angle . Note that P has reflection symmetry iff for some angle . We also know . Hence . Multiply both sides by to have .
(1) Thus P has reflection symmetry iff for some angle .
This simple fact enables a easy method to determine if P has reflection symmetry. Take a simple polygon P. First assume the centroid of P is the origin this is needed to ensure that a rotation about the x-axis is a rotation of P and not a rotation and translation of P.
From (1) we know that for some angle . To determine we pick a vertex, v, of . We know that v is rotated into some vertex w of . Hence we try rotating v into every vertex w of until we get a match.