  Starting from a generalized Hamilton-Jacobi formalism, we develop a new
framework for constructing observables and their evolution in theories
invariant under global time reparametrizations. Our proposal relaxes the usual
Dirac prescription for the observables of a totally constrained system
(`perennials') and allows one to recover the influential partial and complete
observables approach in a particular limit. Difficulties such as the
non-unitary evolution of the complete observables in terms of certain partial
observables are explained as a breakdown of this limit. Identification of our
observables (`mutables') relies upon a physical distinction between gauge
symmetries that exist at the level of histories and states (`Type 1'), and
those that exist at the level of histories and not states (`Type 2'). This
distinction resolves a tension in the literature concerning the physical
interpretation of the partial observables and allows for a richer class of
observables in the quantum theory. There is the potential for the application
of our proposal to the quantization of gravity when understood in terms of the
Shape Dynamics formalism.
