  Comparability graphs are a popular class of graphs. We introduce as the
digraph analogue of comparability graphs the class of comparability digraphs.
We show that many concepts such as implication classes and the knotting graph
for a comparability graph can be naturally extended to a comparability digraph.
We give a characterization of comparability digraphs in terms of their knotting
graphs. Semicomplete comparability digraphs are a prototype of comparability
digraphs. One instrumental technique for analyzing the structure of
comparability graphs is the Triangle Lemma for graphs. We generalize the
Triangle Lemma to semicomplete digraphs. Using the Triangle Lemma for
semicomplete digraphs we prove that if an implication class of a semicomplete
digraph contains no circuit of length 2 then it contains no circuit at all. We
also use it to device an $\mathcal{O}(n^3)$ time recognition algorithm for
semicomplete comparability digraphs where $n$ is the number of vertices of the
input digraph. The correctness of the algorithm implies a characterization for
semicomplete comparability digraphs, akin to that for comparability graphs.
