  We present a novel hierarchical construction of projective spin networks of
the Ponzano-Regge type from an assembling of five quadrangles up to the
combinatorial 4-simplex compatible with a geometrical realization in Euclidean
4-space. The key ingrendients are the projective Desargues configuration and
the incidence structure given by its space-dual, on the one hand, and the
Biedenharn--Elliott identity for the 6j symbol of SU(2), on the other. The
interplay between projective-combinatorial and algebraic features relies on the
recoupling theory of angular momenta, an approach to discrete quantum gravity
models carried out successfully over the last few decades. The role of Regge
symmetry --an intriguing discrete symmetry of the $6j$ which goes beyond the
standard tetrahedral symmetry of this symbol-- will be also discussed in brief
to highlight its role in providing a natural regularization of projective spin
networks that somehow mimics the standard regularization through a
q-deformation of SU(2).
