  Classical background independence is reflected in Lagrangian general
relativity through covariance under the full diffeomorphism group. We show how
this independence can be maintained in a Hamilton-Jacobi approach that does not
accord special privilege to any geometric structure. Intrinsic spacetime
curvature based coordinates grant equal status to all geometric backgrounds.
They play an essential role as a starting point for inequivalent semi-classical
quantizations. The scheme calls into question Wheeler's geometrodynamical
approach and the associated Wheeler-DeWitt equation in which three-metrics are
featured geometrical objects. The formalism deals with variables that are
manifestly invariant under the full diffeomorphism group. Yet, perhaps
paradoxically, the liberty in selecting intrinsic coordinates is precisely as
broad as is the original diffeomorphism freedom. We show how various ideas from
the past five decades concerning the true degrees of freedom of general
relativity can be interpreted in light of this new constrained Hamiltonian
description. In particular, we show how the Kucha\v{r} multi-fingered time
approach can be understood as a means of introducing full four-dimensional
diffeomorphism invariants. Every choice of new phase space variables yields new
Einstein-Hamilton-Jacobi constraining relations, and corresponding intrinsic
Schr\"odinger equations. We show how to implement this freedom by canonical
transformation of the intrinsic Hamiltonian. We also reinterpret and rectify
significant work by B. Dittrich on the construction of `Dirac observables'.
