  We report on an experiment that investigates the spatial mode conversion in
the process of parametric down-conversion seeded by a light beam in a
superposition of orbital angular momentum modes. This process is interpreted in
terms of a geometric representation of first-order spatial modes in a
Poincar\'e sphere, providing an intuitive image of the phase conjugation and
the topological charge conservation. We also make a comparison with the
analogous phenomenon for optical parametric oscillators.
