We construct families of nonisometric hyperbolic orbifolds that contain the same isometry classes of nonflat totally geodesic subspaces. The main tool is a variant of the well-known Sunada method for constructing length-isospectral Riemannian manifolds that handles totally geodesic submanifolds of multiple codimensions simultaneously.
"Constructing geometrically equivalent hyperbolic orbifolds." Algebr. Geom. Topol. 17 (2) 831 - 846, 2017. https://doi.org/10.2140/agt.2017.17.831