  We explore a degenerate four-wave mixing process induced by transversely
structured light beams in a rubidium vapor cell. In particular, we consider the
nonlinear interaction driven by optical modes contained in the orbital angular
momentum Poincar\'e sphere, which can be parametrized in terms of a polar and
an azimuthal angle. In this context we investigate the transfer of spatial
structure to two distinct four-wave mixing signals, possessing different
propagation directions in space. We show that under usual assumptions, the
output fields can also be described by modes belonging to Poincar\'e spheres,
and that the angles describing the input and output modes are related according
to well-defined rules. Our experimental results show good agreement with the
calculations, which predict intricate field structures and a transition of the
FWM transverse profile between the near- and far-field regions.
