Distributive lattice orderings and Priestley duality

The ordering relation of a bounded distributive lattice L is a (distributive) (0, 1)-sublattice of L \times L. This construction gives rise to a functor \Phi from the category of bounded distributive lattices to itself. We examine the interaction of \Phi with Priestley duality and characterise those bounded distributive lattices L such that there is a bounded distributive lattice K such that \Phi(K) is (isomorphic to) L.