  Stochastic symmetries and related invariance properties of finite dimensional
SDEs driven by general cadlag semimartingales taking values in Lie groups are
defined and investigated. The considered set of SDEs, first introduced by S.
Cohen, includes affine and Marcus type SDEs as well as smooth SDEs driven by
Levy processes and iterated random maps. A natural extension to this general
setting of reduction and reconstruction theory for symmetric SDEs is provided.
Our theorems imply as special cases non trivial invariance results concerning a
class of affine iterated random maps as well as symmetries for numerical
schemes (of Euler and Milstein type) for Brownian motion driven SDEs.
