  In gauge theory, it is commonly stated that time-reversal symmetry only
exists at $\theta=0$ or $\pi$ for a $2\pi$-periodic $\theta$-angle. In this
paper, we point out that in both the free Maxwell theory and massive QED, there
is a non-invertible time-reversal symmetry at every rational $\theta$-angle,
i.e., $\theta= \pi p/N$. The non-invertible time-reversal symmetry is
implemented by a conserved, anti-linear operator without an inverse. It is a
composition of the naive time-reversal transformation and a fractional quantum
Hall state. We also find similar non-invertible time-reversal symmetries in
non-Abelian gauge theories, including the $\mathcal{N}=4$ $SU(2)$ super
Yang-Mills theory along the locus $|\tau|=1$ on the conformal manifold.
