Assuming that a fixed orientation has been chosen for 4-dimensional space, isoclinic 4D rotations may be put into two categories. To see this, consider an isoclinic rotation , and take an orientation-consistent ordered set of mutually perpendicular half-lines at (denoted as ) such that and span an invariant plane, and therefore and also span an invariant plane. Now assume that only the rotation angle is specified. Then there are in general four isoclinic rotations in planes and with rotation angle , depending on the rotation senses in and .