One branch of the La-214 family of cuprate superconductors, La1.6−xNd0.4SrxCuO4 (Nd-LSCO), has been of significant and sustained interest, in large part because it displays the full complexity of the phase diagram for canonical hole-doped, high-TC superconductivity, while also displaying relatively low superconducting critical temperatures. The low-superconducting TCs imply that experimentally accessible magnetic fields can suppress the superconductivity to zero temperature. In particular, this has enabled various transport and thermodynamic studies of the T=0 ground state in Nd-LSCO, free of superconductivity, across the critical doping p*=0.23 where the pseudogap phase ends. The strong dependence of its superconducting properties on its crystal symmetry has itself motivated careful studies of the Nd-LSCO structural phase diagram. This paper provides a systematic study and summary of the materials preparation and characterization of both single-crystal and polycrystalline samples of Nd-LSCO. Single-phase polycrystalline samples with x spanning the range from 0.01 to 0.40 have been synthesized, and large single crystals of La1.6−xNd0.4SrxCuO4 for select x across the region (0.07, 0.12, 0.17, 0.19, 0.225, 0.24, and 0.26) were grown by the optical floating-zone method. Systematic neutron and x-ray-diffraction studies on these samples were performed at both low and room temperatures, 10 and 300 K, respectively. These studies allowed us to follow the various structural phase transitions and propose an updated structural phase diagram for Nd-LSCO. In particular, we found that the low-temperature tetragonal (LTT) phase ends at a critical doping pLTT=0.255±0.005, clearly separated from p*.