As compositions of mappings go, the cross ratio commutes with simultaneous projective transformations on each of its arguments. Additionally it commutes with the normal subgroup of simultaneous pair permutation (plus the identity). In the distance interpretation, comparing ratios of distances between one pair of points to another, it is insensitive to exchanging the from points for the to points.
The additional symmetries get special names:
| name | symmetry | ratios |
| harmonic ratio | 1 | |
| anharmonic ratio | -1 | |
| equianharmonic ratio |  |