  We compute twists of the modular curve $X(13)$ that parametrise the elliptic
curves 13-congruent to a given elliptic curve. Searching for rational points on
these twists enables us to find non-trivial pairs of 13-congruent elliptic
curves over ${\mathbb Q}$, i.e. pairs of non-isogenous elliptic curves over
${\mathbb Q}$ whose 13-torsion subgroups are isomorphic as Galois modules. We
also find equations for the surfaces parametrising pairs of 13-congruent
elliptic curves. There are two such surfaces, corresponding to 13-congruences
that do, or do not, respect the Weil pairing. We write each as a double cover
of the projective plane ramified over a highly singular model for Baran's
modular curve of level 13. By finding suitable rational curves on these
surfaces, we show that there are infinitely many non-trivial pairs of
13-congruent elliptic curves over ${\mathbb Q}$.
