Orientation-reversing equivalence of branched-cover monodromy ($P{\rho }_1(a)P^{-1}={\rho }_2(a{)}^{-1},P{\rho }_1(b)P^{-1}={\rho }_2(b{)}^{-1}$)

For covers branched over three real points with a real base point, choose the two generating loops so complex conjugation reverses each. A sheet permutation satisfying the displayed identities lifts that conjugation to an anticonformal equivalence of the covers, including the filled branch points. For closed hyperbolic Riemann surfaces this is a Riemannian isometry. Such an isometry need not come from conjugating the subgroups inside the original deck group, so a Gassmann equivalent pair can be isometric despite nonconjugacy there.