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The halide perovskite heterostructure (CuCl4)2(MTPA)4Cu3Cl6 (Cu_Cu; MTPA = 3-(methylthio)-propylammonium) forms from solution as single crystals consisting of alternating layers of 2D CuII–Cl perovskite and 1D CuII–Cl diamond–chain intergrowth. Using magnetometry, heat capacity, and electron paramagnetic resonance measurements, we interrogate the magnetic ordering of the 2D perovskite and 1D intergrowth layers at temperatures down to 0.055 K. As with other Cu‒Cl perovskites, the perovskite-layer spins order ferromagnetically at 10 K. Magnetization data of Cu_Cu feature a multi–component curve, consistent with magnetization of the perovskite layers and one of the three additional CuII sites in the intergrowth layer, suggesting antiferromagnetic coupling of the remaining two intergrowth-layer spins. A broad feature in AC susceptibility measurements at 6 K and an anomalous heat capacity feature at 0.3 K suggest that local ordering events occur at dramatically different energy scales with decreasing temperature. EPR spectra indicate that these local orderings occur within the 1D chains. Notably, no long–range magnetic ordering event in the intergrowth is evident down to 0.055 K, suggesting that the geometric constraints imposed by the perovskite framework and the steric bulk of the MTPA ligands physically separate and magnetically isolate the diamond chains. In contrast, well–studied diamond-spin-chain materials such as azurite show long-range magnetic order at low-temperatures due to interchain interactions. Thus, Cu_Cu provides an ideal platform for studying isolated, anisotropic spin chains. More generally, this study illustrates the capability of halide perovskite heterostructures to serve as vehicles for the scalable synthesis of complex magnetic materials.