English: Rotations and translations can each be constructed out of exactly two planar reflections, meaning that they are both "bireflections". In the case of a rotation, the reflections are in two planes that are at an angle; in the case of a translation, two parallel planes are needed.

In both cases, the black cube is first reflected by yellow plane to create the yellow cube. The yellow cube is then reflected by the green plane to create the green cube.

In both cases there is one "gauge" aka "degree of freedom". The same rotation will be made by any two planes that meet in the line shown, so long as the planes make the same angle with one another. The same goes for any pair of parallel planes that are the same distance and parallel to the two used to make the translation. In this animation, the planes are shown being moved with the gauge.

In the case of a translation, in a technical sense there still exists an "axis" for the transformation, it is just that the axis is a line at infinity.