The well-known connection between integer matrices with simple spectrum and unit groups in orders of algebraic number fields is used to describe the structure of the toral automorphisms \(F\) that are symmetries of \(G\) (that is, \(FG=GF\)) or are reversing symmetries (that is \(FG=G^{-1}F\)). The rather complete classification available for the 2-torus is extended to the \(n\)-torus here. As the authors point out, the methods used are quite standard in number theory, but are not perhaps widely known in the dynamical systems community.