  Twisted bilayers of high-$T_c$ cuprate superconductors have been argued to
form topological phases with spontaneously broken time reversal symmetry ${\cal
T}$ for certain twist angles. With the goal of helping to identify unambiguous
signatures of these topological phases in transport experiments, we
theoretically investigate a suite of Josephson phenomena between twisted
layers. We find an unusual non-monotonic temperature dependence of the critical
current at intermediate twist angles which we attribute to the unconventional
sign structure of the $d$-wave order parameter. The onset of the ${\cal
T}$-broken phase near $45^\circ$ twist is marked by a crossover from the
conventional $2\pi$-periodic Josephson relation $J(\varphi)\simeq
J_c\sin{\varphi}$ to a $\pi$-periodic function as the single-pair tunneling
becomes dominated by a second order process that involves two Cooper pairs.
Despite this fundamental change, the critical current remains a smooth function
of the twist angle $\theta$ and temperature $T$ implying that a measurement of
$J_c$ alone will not be a litmus test for the ${\cal T}$-broken phase. To
obtain clear signatures of the ${\cal T}$-broken phase one must measure $J_c$
in the presence of an applied magnetic field or radio-frequency drive, where
the resulting Fraunhofer patterns and Shapiro steps are altered in a
characteristic manner. We discuss these results in light of recent experiments
on twisted bilayers of the high-$T_c$ cuprate superconductor
Bi$_2$Sr$_2$CaCu$_2$O$_{8+\delta}$.
