In this paper we investigate an isomorphism σ between a directed de Bruijn digraph B(2,n) and its converse, which is the digraph obtained from B(2,n) by reversing the direction of all its arcs. A cycle C is said to be σ-self-converse when the cycle σ(C) coincides with its converse. We determine a characterization of σ-self-converse cycles, distinguishing the cases of n even and odd. Moreover we prove that, for n even, there does not exist a Hamiltonian σ-self-converse cycle, while, for n odd, we determine a constructive proof of the existence of a similar cycle. Finally we prove that for every n there exists only one σ-self-converse cycle of length 4.