I agree, it's mostly a matter of words. But just for the record, quaternions in general are 4D isoclinic rotations (with optional scaling). Left-multiplication by a unit quaternion corresponds to a left-isoclinic rotation, and right-multiplication corresponds to a right-isoclinic one. By combining them you can produce any 4D rotation. A subset of this is the set of 3D rotations that rotate ix+jy+kz into ix'+jy'+kz'.