Summary: Perfect digraphs have been introduced in [\textit{S. D. Andres} and \textit{W. Hochstättler}, J. Graph Theory 79, No. 1, 21--29 (2015; Zbl 1312.05059)] as those digraphs where, for any induced subdigraph, the dichromatic number and the symmetric clique number are equal. Dually, we introduce a directed version of the clique covering number and define \(\alpha \)-perfect digraphs as those digraphs where, for any induced subdigraph, the clique covering number and the stability number are equal. It is easy to see that \(\alpha \)-perfect digraphs are the complements of perfect digraphs. A digraph is strictly perfect if it is perfect and \(\alpha \)-perfect. We generalise the Strong Perfect Graph Theorem and Lovász asymmetric version of the Weak Perfect Graph Theorem [\textit{L. Lovász}, J. Comb. Theory, Ser. B 13, 95--98 (1972; Zbl 0241.05107)] to the classes of perfect, \( \alpha \)-perfect and strictly perfect digraphs. Furthermore, we characterise strictly perfect digraphs by symmetric chords and non-chords in their directed cycles. As an example for a subclass of strictly perfect digraphs, we show that directed cographs are strictly perfect.