= Orientation-reversing equivalence of branched-cover monodromy
{title2=$P\rho_1(a)P^{-1}=\rho_2(a)^{-1},\quad P\rho_1(b)P^{-1}=\rho_2(b)^{-1}$}
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.
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