the rotoinversion axis
A rotoinversion axis is a two-step combination move: rotate the object by a set angle, then immediately invert it through a point on the axis. Neither step alone is a symmetry of the object, but the pair together is. Think of it as a 'twist-and-flip' done in one breath, the result lands every atom on an identical atom even though a plain rotation would not.
It is written n-bar (a bar over the number) in the international notation: 1-bar, 2-bar, 3-bar, 4-bar, 6-bar. Some of these are old friends in disguise. 1-bar (rotate 360, then invert) is just the inversion centre. 2-bar (rotate 180, then invert) is exactly a mirror plane perpendicular to the axis. 6-bar equals a 3-fold axis with a mirror across it (written 3/m). Only 4-bar and 3-bar bring genuinely new operations you cannot get any simpler way. Rotoinversions are improper operations: they reverse handedness, so any crystal containing one cannot be chiral.
These five rotoinversion axes, together with the five proper rotation axes, are the ten building-block operations from which all 32 crystallographic point groups are assembled. The Schoenflies system prefers a close cousin, the rotoreflection (rotate then reflect, symbol S-n), so the same physical symmetry gets different labels in the two notations, worth remembering when you translate.
A 4-bar axis: rotate 90 degrees, then invert. A tetrahedral molecule like methane has 4-bar axes even though it has no plain 4-fold axis and no inversion centre on its own.
A rotoinversion n-bar rotates by 360/n then inverts, a single improper operation.
2-bar is a mirror and 1-bar is an inversion centre, so the 'new' rotoinversions worth memorising as distinct are really just 3-bar and 4-bar.