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 $2pi$-periodic Josephson relation $J(varphi)simeq J_csin{varphi}$ 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$_{8+delta}$.
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