Large crystals of La2−xCa1+xCu2O6 (La-Ca-2126) with x = 0.10 and 0.15 have been grown and converted to bulk superconductors by high-pressure oxygen annealing. The superconducting transition temperature, Tc, is as high as 55 K; this can be raised to 60 K by post-annealing in air. Here we present structural and magnetic characterizations of these crystals using neutron scattering and muon spin rotation techniques. While the as-grown, non-superconducting crystals are single phase, we find that the superconducting crystals contain 3 phases forming coherent domains stacked along the c axis: the dominant La-Ca-2126 phase, very thin (1.5 unit-cell) intergrowths of La2CuO4, and an antiferromagnetic La8Cu8O20 phase. We propose that the formation and segregation of the latter phases increases the Ca concentration of the La-Ca-2126, thus providing the hole-doping that supports superconductivity.