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 phase and a mirror-protected topological crystalline phase. In this work, we first focus on the naturally occurring mineral called jacutingaite, Pt2HgSe3, and show based on density-functional calculations that it realizes this dual topological phase and that the same conclusion holds for Pd2HgSe3. Second, we introduce tight-binding models that capture the essential topological properties of this dual topological phase in materials with threefold rotation symmetry and use these models to describe the main features of the surface spectral density of different materials in the class.