Reflection intertwining of triangle-cover coset actions ($R{A}_U{R}^{-1}=A_V^{-1},R{B}_U{R}^{-1}=B_V^{-1}$)

Write triangle-group generators as products $a=s_1{s}_2$, $b=s_2{s}_3$ 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.