  For a hyperbolic toral automorphism, we construct a profinite completion of
an isomorphic copy of the homoclinic group of its right action using isomorphic
copies of the periodic data of its left action. The resulting profinite group
has a natural module structure over a ring determined by the right action of
the hyperbolic toral automorphism. This module is an invariant of conjugacy
that provides means in which to characterize when two similar hyperbolic toral
automorphisms are conjugate or not. In particular, this shows for two similar
hyperbolic toral automorphisms with module isomorphic left action periodic
data, that the homoclinic groups of their right actions play the key role in
determining whether or not they are conjugate. This gives a complete set of
dynamically significant invariants for the topological classification of
hyperbolic toral automorphisms.
