  We use a polyhedral criterion for the existence of diagonal splittings to
investigate which toric varieties X are diagonally split. Our results are
stated in terms of the vector configuration given by primitive generators of
the 1-dimensional cones in the fan defining X. We show, in particular, that X
is diagonally split at all q if and only if this configuration is unimodular,
and X is not diagonally split at any q if this configuration is not 2-regular.
We also study implications for the possibilities for the set of q at which a
toric variety X is diagonally split.
