A rigid ellipsoidal inclusion is embedded at arbitrary orientation in a homogeneous, arbitrarily anisotropic, elastic matrix and is rotated infinitesimally by means of an imposed couple. Far away the matrix remains either unstrained or at a prescribed uniform strain. A simple `singularity' representation of the elastic field is proposed. It yields directly the relation between the couple and rotation vectors, and the stress, strain and rotation concentrations over the ellipsoidal surface, without having to solve either the governing equations of equilibrium in the matrix, or the fundamental ones of a point force. A full solution is given for an isotropic matrix.