Many normalizations in various classes of linear dynamical state-space systems lead to system representations which are determined up to a state isometry. Here we present a new set of techniques to obtain (local) canonical forms under state isometries, using what we call sub-diagonal pivot structures. These techniques lead to a very flexible, straightforward algorithm to put any system into canonical form under state isometries. The parametrization of these canonical forms is discussed for a number of classes, including lossless systems and input-normal stable systems both in discrete time and in continuous time.