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Bourel, Dickenstein and Rittatore characterized (in 2011) the self-dual projective toric varieties $X_A\subset {\textbf P}(V)$ equivariantly embedded in terms of the combinatorics of the associated configuration of weights and also in terms of the geometry of the action of the torus. In an ongoing joint work with Apostolos Thoma we complete their classification by describing all of the matrices $A$ such that $X_A$ is a self-dual projective toric variety. The aim of this talk is to explain this result. The main ingredient needed for this is a combinatorial classification of all toric ideals given by Sonja Petrovic, Apostolos Thoma and myself in 2015. Self-dual varieties are a special case of defective varieties, and the complete classification of defective projective toric varieties in an equivariant embedding is open in full generality.