We classify the countably infinite oriented graphs which, for
partition of their vertex set into two parts, induce an isomorphic copy of
themselves on at least one of the parts. These graphs are the edgeless graph,
the random tournament, the transitive tournaments of order
type~$\omega^\alpha$, and two orientations of the Rado graph: the random
oriented graph, and a newly found random acyclic oriented graph.
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