  We examine several retro-reflecting optical elements, each involving two
reflections. In the case of a hollow metallic cone having an apex angle of
90\degree, a circularly-polarized incident beam acquires, upon reflection, the
opposite spin angular momentum. However, no angular momentum is transferred to
the cone, because the reflected beam picks up an orbital angular momentum that
is twice as large but opposite in direction to that of its spin. A 90\degree
cone made of a transparent material in which the incident light suffers two
total internal reflections before returning, may be designed to endow the
retro-reflected beam with different mixtures of orbital and spin angular
momenta. Under no circumstances, however, is it possible to transfer angular
momentum from the light beam to the cone without either allowing absorption or
breaking the axial symmetry of the cone. A simple example of broken symmetry is
provided by a wedge-shaped metallic reflector having an apex angle of
90\degree, which picks up angular momentum upon reflecting a
circularly-polarized incident beam.
