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We have studied a large class of low-dimensional quantum magnets based on S = 1 Ni(II) ions. When coordinated to varying numbers of organic ligands and halides, we can observe all three universality classes; Heisenberg, Ising and XY. We show that the trend in the magnitude of the single-ion anisotropy (D) coincides with predictions from the spectrochemical series. Also, it is found that NiX₂N₄ coordination spheres (X = Br, I, NCS) have D < 0 (Ising-like) whereas X = Cl and F exhibit ground states with large, positive D (XY-like). Increasing the number of halide or oxygen donor atoms around the Ni(II) ion, to form NiX₄N₂ or NiO₄N₂, induces an Ising-like ground state in the existing examples. In the absence of frustrated interactions, we demonstrate an ability to predict the orientation of the Ni(II) spin direction in simple systems, regardless of dimensionality or presence of long-range magnetic order, based solely on the single-ion anisotropy.