  We define the Z/2Z Hodge spaces H_{pq}(\Sigma) of a fan \Sigma.
  If \Sigma is the normal fan of a reflexive polytope \Delta then we use
polyhedral duality to compute the Z/2Z Hodge Spaces of \Sigma. In particular,
if the cones of dimension at most e in the face fan \Sigma^* of \Delta are
smooth then we compute H_{pq}(\Sigma) for p<e-1. If \Sigma^* is a smooth fan
then we completely determine the spaces H_{pq}(\Sigma) and we show the toric
variety X associated to \Sigma is maximal, meaning that the sum of the Z/2Z
Betti numbers of X(R) is equal to the sum of the Z/2Z Betti numbers of X(C).
