• antisymmetry;
  • double antisymmetry space groups;
  • rotation-reversal

Rotation-reversal symmetry was recently introduced to generalize the symmetry classification of rigid static rotations in crystals such as tilted octahedra in perovskite structures and tilted tetrahedra in silica structures. This operation has important implications for crystallographic group theory, namely that new symmetry groups are necessary to properly describe observations of rotation-reversal symmetry in crystals. When both rotation-reversal symmetry and time-reversal symmetry are considered in conjunction with space-group symmetry, it is found that there are 17 803 types of symmetry which a crystal structure can exhibit. These symmetry groups have the potential to advance understanding of polyhedral rotations in crystals, the magnetic structure of crystals and the coupling thereof. The full listing of the double antisymmetry space groups can be found in the supplementary materials of the present work and at