This communication compiles propositions concerning the spherical geometry of rotations when represented by unit quaternions. The propositions are thought to establish a two-way correspondence between geometrical objects like circle, sphere, and torus spanned by real unit quaternions representing rotations and geometrical objects constituted by directions and their image when subjected to these rotations. Our geometrical approach leads to an alternative proof of a spherical variant of Asgeirsson's mean value theorem interpreted as equivalence of a certain integration in the direct or inverse pole figure, and to clarifications of the interpretation of some integrals related to the spherical Radon transform of orientation probability density functions of unit quaternions and their inverse.