Two-dimensional topological insulators show great promise for spintronic applications. Much attention has been placed on single atomic or molecular layers, such as bismuthene. The selections of such materials are, however, limited. To broaden the base of candidate materials with desirable properties for applications, we report herein an exploration of the physics of double layers of bismuthene and antimonene. The electronic structure of a film depends on the number of layers, and it can be modified by epitaxial strain, by changing the effective spin-orbit coupling strength, and by the manner in which the layers are geometrically stacked. First-principles calculations for the double layers reveal a number of phases, including topological insulators, topological semimetals, Dirac semimetals, trivial semimetals, and trivial insulators. Their phase boundaries and the stability of the phases are investigated. The results illustrate a rich pattern of phases that can be realized by tuning lattice strain and effective spin-orbit coupling.