Predicting the full spin Hall conductivity tensor of a magnetic crystal is difficult because crystallographic and magnetic symmetries do not constrain all parts of the response in the same way. Here, we separate the spin Hall conductivity of altermagnets into time-reversal-even and time-reversal-odd channels and combine symmetry analysis with first-principles calculations for six representative compounds spanning distinct crystallographic classes. The even channel follows crystallographic selection rules consistent with its Berry-curvature Fermi-sea origin. The odd channel is set by magnetic symmetry and can be reshaped by antiunitary operations whose spatial parts originate from screw or glide symmetries. Momentum-resolved analysis further links the even response to spin-orbit-driven avoided crossings and the odd response to anisotropic near-Fermi-level spin-current response textures. By converting crystallographic and magnetic symmetry operations into linear constraints on the full 27-component SHC tensor, this framework provides a computational prescreening route for identifying symmetry-allowed spin Hall channels before dense first-principles transport calculations. Extending the same analysis to 62 spin-split collinear antiferromagnets yields a magnetic-point-group tensor atlas, identifying design limits such as the complete 13/14tensor partition in CuF2 and the one-parameter time-odd tensor compression in MnSe.