We begin a systematic study of Euclidean piecewise isometric dynamical systems (p.i.d.s.) with a particular focus on the interplay between geometry, symbolic dynamics, and the group of isometries associated with p.i.d.s. We investigate various aspects of the dynamical information contained in the coding: symbolic
growth and the periodic behavior of codings and cells. This theoretical investigation is motivated by the many examples of piecewise isometric dynamical systems found recently in the literature. Piecewise isometric dynamical systems are direct generalizations of interval exchange transformations to non-invertible, higher dimensional maps.