  Antiferromagnetic materials (AFMs) have been gaining lots of attentions due
to its great potential in spintronics devices and the recently discovered novel
spin structure in the momentum space, i.e., $C$-paired spin-valley or
spin-momentum locking (CSVL), where spins and valleys/momenta are locked to
each other due to the crystal symmetry guaranteeing zero magnetization. Here,
we systematically studied CSVLs and proposed a general theory and algorithm
using little co-group and coset representatives, which reveals that 12
elementary kinds of CSVLs, determined by the geometric relation of spins and
valleys and the essential symmetry guaranteeing zero magnetization, are
sufficient to fully represent all possible CSVLs. By combining the proposed
algorithm and high-throughput first-principles calculations, we predicted 38
magnetic point groups and identified 140 experimentally verified AFMs that can
realize CSVL. Besides predicting new materials, our theory can naturally reveal
underlying mechanisms of CSVLs' responses to external fields. As an example,
two qualitatively different types of piezomagnetism via occupation imbalance or
spin tilting were predicted in RuO$_2$. The algorithm and conclusions can be
directly extended to the locking between valley/momentum and any other
pseudo-vector degree of freedom, e.g. Berry curvature, as exemplified in
RuO$_2$, and the proposed concept and methodology can be straightforwardly
applied to other symmetry groups such as spin space group.
