The orthogonal group, consisting of all proper and improper rotations, is generated by reflections.

For an improper rotation, v does not in general even keep the same magnitude:

A length-preserving transformation which reverses orientation is called an improper rotation.

However, if improper rotations are also taken into consideration, then the spin space and its dual are not isomorphic.

In crystallography, an overline indicates an improper rotation or a negative number:

The 4th and 5th in particular, and in a wider sense the 6th also, are called improper rotations.

An improper rotation of an object thus produces a rotation of its mirror image.

An improper rotation can be understood as an inversion followed by a proper rotation.

Then, any orthogonal matrix is either a rotation or an improper rotation.

When 1, the matrix is an improper rotation.