Effects of Roto-Inversion
Only a small number of roto-inversion axes are needed. Many of them actually reduce to other symmetry elements. Remember, roto-inversion consists of a rotation, followed by inverting through a point. The effects on coordinates are shown for each one below.
Two-fold roto-inversion
For cartesian axes: $$(x, y, z) \rightarrow (-x, -y, z) \rightarrow (x, y, -z)$$
This is identical to a mirror plane.
Three-fold roto-inversion
For hexagonal axes: $$(x, y, z) \rightarrow (-y, x-y, z) \rightarrow (y, y-x, -z)$$
While the rotation and inversion is supposed to be applied in one step, three-fold roto-inversions are actually identical to the separate presence of a three-fold rotation and an inversion centre (\(\bar{1}\)).
Four-fold roto-inversion
For cartesian axes:
$$(x, y, z) \rightarrow (-y, x, z) \rightarrow (y, -x, -z)$$
Four-fold roto-inversions are a new symmetry element.
Six-fold roto-inversion
For hexagonal axes:
$$(x, y, z) \rightarrow (-y, x-y, z) \rightarrow (y, y-x, -z)$$
Six-fold roto-inversions are actually equivalent to the separate application of a three-fold rotation and a perpendicular mirror plane (\(\frac{3}{m}\)). The space group \(P\frac{3}{m}\) is not listed in The International Tables as it is isomorphic with \(P\bar{6}\)!