Conjugacy classes of the symmetry group of a square ($\{e\},\{r^2\},\{r,r^3\},\{s,r^{2}s\},\{rs,r^{3}s\}$)

The displayed conjugacy classes have sizes one, one, two, two and two. The identity and half-turn are central elements. Reflection sends $r$ to $r^{-1}$, pairing the two quarter-turns. Conjugating a reflection by $r$ sends $r^{k}s$ to $r^{k+2}s$; conjugating by $s$ sends it to $r^{-k}s$. Thus the parity of $k$ is invariant under conjugation, separating the axis reflections from the diagonal reflections, and each displayed pair is one conjugacy class.