  Clean isotropic quantum Hall fluids in the continuum possess a host of
symmetry-protected quantized invariants, such as the Hall conductivity, shift
and Hall viscosity. Here we develop a theory of symmetry-protected quantized
invariants for topological phases defined on a lattice, where quantized
invariants with no continuum analog can arise. We develop topological field
theories using discrete crystalline gauge fields to fully characterize
quantized invariants of (2+1)D Abelian topological orders with symmetry group
$G = U(1) \times G_{\text{space}}$, where $G_{\text{space}}$ consists of
orientation-preserving space group symmetries on the lattice. We show how
discrete rotational and translational symmetry fractionalization can be
characterized by a discrete spin vector, a discrete torsion vector which has no
analog in the continuum or in the absence of lattice rotation symmetry, and an
area vector, which also has no analog in the continuum. The discrete torsion
vector implies a type of crystal momentum fractionalization that is only
non-trivial for $2$, $3$, and $4$-fold rotation symmetry. The quantized
topological response theory includes a discrete version of the shift, which
binds fractional charge to disclinations and corners, a fractionally quantized
angular momentum of disclinations, rotationally symmetric fractional charge
polarization and its angular momentum counterpart, constraints on charge and
angular momentum per unit cell, and quantized momentum bound to dislocations
and units of area. The fractionally quantized charge polarization, which is
non-trivial only on a lattice with $2$, $3$, and $4$-fold rotation symmetry,
implies a fractional charge bound to lattice dislocations and a fractional
charge per unit length along the boundary. An important role is played by a
finite group grading on Burgers vectors, which depends on the point group
symmetry of the lattice.
