(Communicated by Bernold Fiedler) Abstract. We study various aspects of the dynamics induced by integer ma-trices on the invariant rational lattices of the torus in dimension 2 and greater. Firstly, we investigate the orbit structure when the toral endomorphism is not invertible on the lattice, characterising the pretails of eventually periodic or-bits. Next we study the nature of the symmetries and reversing symmetries of toral automorphisms on a given lattice, which has particular relevance to (quantum) cat maps.