The large sample theory of spherical regression with fixed explanatory variables (Chang, 1986) is modified for the case of axes, unsigned directional data. Thus, for the case in which u1, . . ., un are fixed points on the sphere with antipodes identified, and v1, . . ., vn are random points such that the distribution of vi depends only on (viT Aui)2 for some unknown rotation matrix A, this paper provides asymptotic tests and confidence regions for A, and for the axis of rotation of A. Results are given in arbitrary dimension.