  We introduce a definition of symmetry generating vector fields on manifolds
which are equipped with a first-order reductive Cartan geometry. We apply this
definition to a number of physically motivated examples and show that our newly
introduced notion of symmetry agrees with the usual notions of symmetry of
affine, Riemann-Cartan, Riemannian and Weizenb\"ock geometries, which are
conventionally used as spacetime models. Further, we discuss the case of Cartan
geometries which can be used to model observer space instead of spacetime. We
show which vector fields on an observer space can be interpreted as symmetry
generators of an underlying spacetime manifold, and may hence be called
"spatio-temporal". We finally apply this construction to Finsler spacetimes and
show that symmetry generating vector fields on a Finsler spacetime are indeed
in a one-to-one correspondence with spatio-temporal vector fields on its
observer space.
