For a fixed digraph H, the minimum cost homomorphism problem, MinHOM(H), asks whether an input digraph G, with given costs c i (u), u∈ ∈V(G), i∈ ∈V(H), and an integer k, admits a homomorphism to H of total cost not exceeding k. Minimum cost homomorphism problems encompass many well studied optimization problems such as list homomorphism problems, retraction and precolouring extension problems, chromatic partition optimization, and applied problems in repair analysis. For undirected graphs the complexity of the problem, as a function of the parameter H, is well understood; for digraphs, the situation appears to be more complex, and only partial results are known. We focus on the minimum cost homomorphism problem for reflexive digraphs H. It is known that MinHOM(H) is polynomial if H has a Min-Max ordering. We prove that for any other reflexive digraph H, the problem MinHOM(H) is NP-complete. (This was earlier conjectured by Gutin and Kim.) Apart from undirected graphs, this is the first general class of digraphs for which such a dichotomy has been proved. Our proof involves a forbidden induced subgraph characterization of reflexive digraphs with a Min-Max ordering, and implies a polynomial test for the existence of a Min-Max ordering in a reflexive digraph H.