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Motivated by the recent proposals for unconventional emergent physics in twisted bilayers of nodal superconductors, we study the peculiarities of the Josephson effect at the twisted interface between d-wave superconductors. We demonstrate that for clean interfaces with a twist angle θ0 in the range 0◦ < θ0 < 45◦, the critical current can exhibit nonmonotonic temperature dependence with a maximum at a nonzero temperature as well as a complex dependence on the twist angle at low temperatures. These effects are shown to reflect the destructive interference between the d-wave order parameters near the nodes at nonzero twist angle. Close to 45◦ we find that the critical current does not vanish due to Cooper pair cotunneling, which can lead to the transition to a time-reversal breaking superconducting d + id phase, which can be suppressed by the interface roughness. We provide a comprehensive theoretical analysis of experiments that can reveal this cotunneling for twisted superconductors close to θ0 = 45◦. In particular, we demonstrate that both the emergence of the Fraunhofer interference pattern near θ0 = 45◦ and fractional Shapiro steps yield unambiguous evidence of Cooper pair cotunneling, necessary for topological superconductivity.