Improper rotation

In geometry, an improper rotation (also called rotation-reflection, rotoreflection, rotary reflection, or rotoinversion) is an isometry in Euclidean space that is a combination of a rotation about an axis and a reflection in a plane perpendicular to that axis. Reflection and inversion are each a special case of improper rotation. Any improper rotation is an affine transformation and, in cases that keep the coordinate origin fixed, a linear transformation. It is used as a symmetry operation in the context of geometric symmetry, molecular symmetry and crystallography, where an object that is unchanged by a combination of rotation and reflection is said to have improper rotation symmetry.

Example polyhedra with rotoreflection symmetry
Group S4 S6 S8 S10 S12
Subgroups C2 C3, S2 = Ci C4, C2 C5, S2 = Ci C6, S4, C3, C2
Example
beveled digonal antiprism
triangular antiprism
square antiprism
pentagonal antiprism
hexagonal antiprism
Antiprisms with directed edges have rotoreflection symmetry.
p-antiprisms for odd p contain inversion symmetry, Ci.
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