  We describe explicitly all multisets of weights whose defining projective
toric varieties are self-dual. In addition, we describe a remarkable and
unexpected combinatorial behaviour of the defining ideals of these varieties.
The toric ideal of a self-dual projective variety is weakly robust, that means
the Graver basis is the union of all minimal binomial generating sets. When, in
addition, the self-dual projective variety has a non-pyramidal configuration,
then the toric ideal is strongly robust, namely the Graver basis is a minimal
generating set, therefore there is only one minimal binomial generating set
which is also a reduced Gr\"obner basis with respect to every monomial order
and thus, equals the universal Gr\"obner basis.
