  We present a technique for the enumeration of all isotopically distinct ways
of tiling a hyperbolic surface of finite genus, possibly nonorientable and with
punctures and boundary. This provides a generalization of the enumeration of
Delaney-Dress combinatorial tiling theory on the basis of isotopic tiling
theory. To accomplish this, we derive representations of the mapping class
group of the orbifold associated to the symmetry group in the group of outer
automorphisms of the symmetry group of a tiling. We explicitly give
descriptions of certain subgroups of mapping class groups and of tilings as
decorations on orbifolds, namely those that are commensurate with the
Primitive, Diamond and Gyroid triply-periodic minimal surfaces. We use this
explicit description to give an array of examples of isotopically distinct
tilings of the hyperbolic plane with symmetries generated by rotations,
outlining how the approach yields an unambiguous enumeration.
