  Topological phases of electronic systems often coexist in a material,
well-known examples being systems which are both strong and weak topological
insulators. More recently, a number of materials have been found to have the
topological structure of both a weak topological insulator and a
mirror-protected topological crystalline insulator. In this work, we first
focus on the naturally occurring mineral called Jacutingaite, Pt$_2$HgSe$_3$,
and show based on density-functional calculations that it realizes this dual
topological phase and that the same conclusion holds for Pd$_2$HgSe$_3$.
Second, we introduce tight-binding models that capture the essential
topological properties of this dual topological phase in materials with
three-fold rotation symmetry and use these models to describe the main features
of the surface spectral density of different materials in the class.
