Spin point groups classify magnetic phases in the weak spin-orbit coupling regime and characterize the static properties of altermagnetic phases, but their dynamical consequences remain largely unexplored. Here, we derive selection rules for high-harmonic generation of charge and spin currents by extending dynamical symmetry to include spin point group operations. Since spin currents transform under both real and spin space operations, whereas charge currents transform only under real space operations, spin current selection rules can reveal magnetic information that is inaccessible to charge current harmonics. In a minimal altermagnetic model, an axis-aligned linearly polarized drive is non-diagnostic for distinguishing ferromagnetic and altermagnetic phases, although the antiferromagnetic phase is distinguished by the absence of the corresponding spin-current harmonics. A diagonal linearly polarized drive distinguishes the three spin-point-group phases within the weak-coupling spin-group description, whereas a single-helicity circularly polarized drive provides a sharper spin-current-harmonic criterion for distinguishing them from magnetic-point-group mimics. These results establish spin current harmonics as a dynamical probe of spin group symmetry. Finite spin-orbit coupling lifts these zeros only perturbatively: the forbidden harmonics acquire amplitudes analytic in the coupling strength, so that the fingerprint becomes a small same-order helicity contrast, below $3\times {10}^{-2}$ at low orders for couplings up to ${10}^{-2}$ of the hopping in our model.