The concept of spin symmetries has gained renewed interest as a valuable tool for classifying unconventional magnetic phases, including altermagnets and recently identified 𝑝-wave magnets. In this work, we show that in compounds with weak spin-orbit coupling, the dominant spin and charge photoresponse is determined by spin group rather than the conventional magnetic group symmetry. As a concrete realization we consider the nonlinear shift photocurrent in Mn5⁢Si3, a material that features the two possible classes of unconventional 𝑝-wave magnetism in the form of two competing spin structures, a coplanar and noncoplanar one. While both are predicted to generate shift currents based on magnetic symmetry considerations, only the noncoplanar configuration survives the spin symmetry requirements. This is numerically confirmed by our ab initio calculations, providing a protocol to experimentally identify the spin configuration of this promising material in photogalvanic or transport measurements.