  Motivated by the wealth of proposals and realizations of nontrivial
topological phases in EuCd$_2$As$_2$, such as a Weyl semimetallic state and the
recently discussed semimetallic versus semiconductor behavior in this system,
we analyze in this work the role of the delicate interplay of Eu magnetism,
strain and pressure on the realization of such phases. For that we invoke a
combination of a group theoretical analysis with ab initio density functional
theory calculations and uncover a rich phase diagram with various non-trivial
topological phases beyond a Weyl semimetallic state, such as axion and
topological crystalline insulating phases, and discuss their realization.
