  For over a century, the structure of materials has been described by a
combination of rotations, rotation-inversions and translational symmetries. By
recognizing the reversal of static structural rotations between clockwise and
counterclockwise directions as a distinct symmetry operation, here we show that
there are many more structural symmetries than are currently recognized in
right- or left-handed handed helices, spirals, and in antidistorted structures
composed equally of rotations of both handedness. For example, though a helix
or spiral cannot possess conventional mirror or inversion symmetries, they can
possess them in combination with the rotation reversal symmetry. Similarly, we
show that many antidistorted perovskites possess twice the number of symmetry
elements as conventionally identified. These new symmetries predict new forms
for "roto" properties that relate to static rotations, such as rotoelectricity,
piezorotation, and rotomagnetism. They also enable symmetry-based search for
new phenomena, such as multiferroicity involving a coupling of spins, electric
polarization and static rotations. This work is relevant to structure-property
relationships in all material structures with static rotations such as
minerals, polymers, proteins, and engineered structures.
