= Reflection intertwining of triangle-cover coset actions
{title2=$R A_U R^{-1}=A_V^{-1},\quad R B_U R^{-1}=B_V^{-1}$}
Write triangle-group generators as products $a=s_1s_2$, $b=s_2s_3$ of adjacent side reflections. Conjugation by $s_2$ sends both to their inverses. A sheet relabelling satisfying the displayed two permutation identities therefore lifts this base reflection to an <orientation-reversing isometry> between the two hyperbolic covers. Thus nonconjugate subgroups in the orientation-preserving finite group can still give isometric covers.
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