  In its canonical formulation, general relativity is subject to gauge
transformations that are equivalent to space-time coordinate changes of general
covariance only when the gauge generators, given by the Hamiltonian and
diffeomorphism constraints, vanish. Since the specific form taken by Poisson
brackets of the constraints and of the gauge transformations and equations of
motion they generate is important for general covariance to be realized,
modifications of the canonical theory, suggested for instance by approaches to
quantum gravity, are not guaranteed to be compatible with the existence of a
covariant space-time line element. This caveat applies even if the modification
preserves the number of independent gauge transformations and the modified
constraints remain first class. Here, a complete derivation of covariance
conditions, regained from the canonical constraints without assuming that
space-time has its classical structure, is presented and applied in detail to
spherically symmetric vacuum models. As a broad application, the presence of
structure functions in the constraint brackets plays a crucial role, which in
an independent analysis has recently been shown to lead to higher algebraic
structures in hypersurface deformations given by an L-infinity bracket. The
physical analysis of a related feature presented here demonstrates that, at
least within the spherically symmetric setting, new theories of modified
gravity are possible that are not of higher-curvature form.
