  In this paper, we completely characterize time-reversal invariant
three-dimensional Chern-Simons gauge theories with torus gauge group. At the
level of the Lagrangian, toral Chern-Simons theory is defined by an integral
lattice, while at the quantum level, it is entirely determined by a quadratic
function on a finite Abelian group and an integer mod 24. We find that quantum
time-reversally symmetric theories can be defined by classical Lagrangians
defined by integral lattices which have self-perpendicular embeddings into a
unimodular lattice. We find that the quantum toral Chern-Simons theory admits a
time-reversal symmetry iff the higher Gauss sums of the associated modular
tensor category are real. We conjecture that the reality of the higher Gauss
sums is necessary and sufficient for a general non-Abelian Chern-Simons to
admit quantum T-symmetry.
