A prominent signature of Majorana bound states is the exotic Josephson effects they produce, the classic example being a fractional Josephson current with 4π periodicity in the phase difference across the junction. Recent work established that topological insulator edges support a novel 'magneto- Josephson effect', whereby a dissipationless current exhibits 4π-periodic dependence also on the relative orientation of the Zeeman fields in the two banks of the junction. Here, we explore the magneto-Josephson effect in junctions based on spin–orbit-coupled quantum wires. In contrast to the topological insulator case, the periodicities of the magneto-Josephson effect no longer follow from an exact superconductor–magnetism duality of the Hamiltonian. We employ numerical calculations as well as analytical arguments to identify the domain configurations that display exotic Josephson physics for quantum-wire junctions, and elucidate the characteristic differences with the corresponding setups for topological insulators edges. To provide guidance to experiments, we also estimate the magnitude of the magneto-Josephson effects in realistic parameter regimes, and compare the Majorana-related contribution to the coexisting 2π-periodic effects emerging from non-Majorana states.