  We study covariant models for vacuum spherical gravity within a canonical
setting. Starting from a general ansatz, we derive the most general family of
Hamiltonian constraints that are quadratic in first-order and linear in
second-order spatial derivatives of the triad variables, and obey certain
specific covariance conditions. These conditions ensure that the dynamics
generated by such family univocally defines a spacetime geometry, independently
of gauge or coordinates choices. This analysis generalizes the Hamiltonian
constraint of general relativity, though keeping intact the covariance of the
theory, and leads to a rich variety of new geometries. We find that the
resulting geometries depend on seven free functions of one scalar variable, and
we study their generic features. By construction, there are no propagating
degrees of freedom in the theory. However, we also show that it is possible to
add matter to the system by simply following the usual minimal-coupling
prescription, which leads to novel models to describe dynamical scenarios.
