Orbit-separating invariant for the standard dihedral action on the Riemann sphere
= Orbit-separating invariant for the standard dihedral action on the Riemann sphere
For $r(z)=\zeta z$ and $s(z)=1/z$, where $\zeta^n=1$, the rational function
$$
z^n+z^{-n}
$$
is invariant under $D_{2n}$. Equality of two values factors as $(u-v)(uv-1)=0$ for $u=z_1^n$ and $v=z_2^n$, so every fibre is exactly one dihedral orbit.