Many nice properties of toric varieties can be deduced from the fact that they admit a splitting of the Frobenius morphism. Varieties whose product with themselves admits a splitting compatible with the diagonal exhibit even nicer properties. For example, every ample line bundle on such a variety is normally generated. In this talk I will give an overview over the consequences of the existence of such splittings and discuss new combinatorial criteria for toric varieties to be diagonally split.