Abstract In this paper we introduce the class of adamant digraphs. These are the digraphs with the property that for any two vertices x and y, the set of successors of x and the set of successors of y are either disjoint or (inclusionwise) comparable. Those adamant digraphs whose inverse digraph is also adamant are called inflexible. This subclass includes many previously known classes, e.g. minimal series-parallel digraphs and Ferrers digraphs. For both adamant and inflexible digraphs we give alternative characterizations and linear-time recognition algorithms. The special case of symmetric adamant digraphs is investigated.