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Symmetry invariants of a group specify the classes of quasiparticles, namelythe classes of projective irreducible \emph{co-representations} (irReps) insystems having that symmetry. More symmetry invariants exist in discrete pointgroups than the full rotation group $\mathrm{O(3)}$, leading to newquasiparticles restricted to lattices that have not any counterpart in vacuum.We focus on the fermionic quasiparticle excitations under ``spin-space group''symmetries, applicable to materials where long-range magnetic structure anditinerant electrons coexist. We provide a list of 218 classes of newquasiparticles with symmetry group $P\times Z_2^T$ (with $P$ a crystallographicpoint group) that can \emph{only} be realized in the spin-space groups, andthen prove a theorem that each of these new types of quasiparticles has uniquephysical properties in terms of the dispersion and the coupling to externalprobe fields.

Symmetry invariants and classes of quasi-particles in magnetically ordered systems having weak spin-orbit coupling