  We discuss the relation between spacetime diffeomorphisms and gauge
transformations in theories of the Yang-Mills type coupled with Einstein's
General Relativity. We show that local symmetries of the Hamiltonian and
Lagrangian formalisms of these generally covariant gauge systems are equivalent
when gauge transformations are required to induce transformations which are
projectable under the Legendre map. Although pure Yang-Mills gauge
transformations are projectable by themselves, diffeomorphisms are not. Instead
the projectable symmetry group arises from infinitesimal
diffeomorphism-inducing transformations which must depend on the lapse function
and shift vector of the spacetime metric plus associated gauge transformations.
Our results are generalizations of earlier results by ourselves and by
Salisbury and Sundermeyer.
