Min-Max orderings correspond to conservative lattice polymorphisms. Digraphs with Min-Max orderings have polynomial time solvable minimum cost homomorphism problems. They can also be viewed as digraph analogues of proper interval graphs and bigraphs. We give a forbidden structure characterization of digraphs with a Min-Max ordering which implies a polynomial time recognition algorithm. We also similarly characterize digraphs with an extended Min-Max ordering, and we apply this characterization to prove a conjectured form of dichotomy for minimum cost homomorphism problems.