Abstract: It is well known that in a generally covariant gravitational theory the choice of spacetime scalars as coordinates yields phase-space observables (or "invariants"). However, their relation to the symmetry group of diffeomorphism transformations has remained obscure. In a symmetry-inspired approach we construct invariants (observables) out of canonically induced active gauge transformations. We make full contact with the "evolving constants of motion" program. All invariants can be obtained as limits of a family of canonical transformations. This permits a short proof that the invariants satisfy Poisson brackets that are equal to the invariants of their corresponding Dirac brackets.