  We study continuum quantum field theories in 2+1 dimensions with
time-reversal symmetry $\cal T$. The standard relation ${\cal T}^2=(-1)^F$ is
satisfied on all the "perturbative operators" i.e. polynomials in the
fundamental fields and their derivatives. However, we find that it is often the
case that acting on more complicated operators ${\cal T}^2=(-1)^F {\cal M}$
with $\cal M$ a non-trivial global symmetry. For example, acting on monopole
operators, $\cal M$ could be $\pm1$ depending on the magnetic charge. We study
in detail $U(1)$ gauge theories with fermions of various charges. Such a
modification of the time-reversal algebra happens when the number of odd charge
fermions is $2 ~{\rm mod}~4$, e.g. in QED with two fermions. Our work also
clarifies the dynamics of QED with fermions of higher charges. In particular,
we argue that the long-distance behavior of QED with a single fermion of charge
$2$ is a free theory consisting of a Dirac fermion and a decoupled topological
quantum field theory. The extension to an arbitrary even charge is
straightforward. The generalization of these abelian theories to $SO(N)$ gauge
theories with fermions in the vector or in two-index tensor representations
leads to new results and new consistency conditions on previously suggested
scenarios for the dynamics of these theories. Among these new results is a
surprising non-abelian symmetry involving time-reversal.
