  In this paper we investigate the canonical structure of diffeomorphism
invariant phase spaces for spatially locally homogeneous spacetimes with
3-dimensional compact closed spaces. After giving a general algorithm to
express the diffeomorphism-invariant phase space and the canonical structure of
a locally homogeneous system in terms of those of a homogeneous system on a
covering space and a moduli space, we completely determine the canonical
structures and the Hamiltonians of locally homogeneous pure gravity systems on
orientable compact closed 3-spaces of the Thurston-type $E^3$, $\Nil$ and
$\Sol$ for all possible space topologies and invariance groups. We point out
that in many cases the canonical structure becomes degenerate in the moduli
sectors, which implies that the locally homogeneous systems are not canonically
closed in general in the full diffeomorphism-invariant phase space of generic
spacetimes with compact closed spaces.
