Twin-width and permutations

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.