  A relationship between the tetrahedron equation for maps and the consistency
property of integrable discrete equations on $\mathbb{Z}^3$ is investigated.
Our approach is a generalization of a method developed in the context of
Yang-Baxter maps, based on the invariants of symmetry groups of the lattice
equations. The method is demonstrated by a case-by-case analysis of the
octahedron type lattice equations classified recently, leading to some new
examples of tetrahedron maps and integrable coupled lattice equations.
