  We prove that a regular category $\mathcal C$ is a Mal'tsev category if and
only if a strong form of the denormalised $3 \times 3$ Lemma holds true in
$\mathcal C$. In this version of the $3 \times 3$ Lemma, the vertical exact
forks are replaced by pullbacks of regular epimorphisms along arbitrary
morphisms. The shape of the diagram it determines suggests to call it the
Cuboid Lemma. This new characterisation of regular categories that are Mal'tsev
categories (= $2$-permutable) is similar to the one previously obtained for
Goursat categories (= $3$-permutable). We also analyse the "relative" version
of the Cuboid Lemma and extend our results to that context.
