  Inspired by a width invariant defined on permutations by Guillemot and Marx,
the twin-width invariant has been recently introduced by Bonnet, Kim,
Thomass\'e, and Watrigant. We prove that a class of binary relational
structures (that is: edge-colored partially directed graphs) has bounded
twin-width if and only if it is a first-order transduction of a~proper
permutation class. As a by-product, it shows that every class with bounded
twin-width contains at most $2^{O(n)}$ pairwise non-isomorphic $n$-vertex
graphs.
