  We give a complete characterization of pseudovarieties of semigroups whose
finitely generated relatively free profinite semigroups are equidivisible.
Besides the pseudovarieties of completely simple semigroups, they are precisely
the pseudovarieties that are closed under Mal'cev product on the left by the
pseudovariety of locally trivial semigroups. A further characterization which
turns out to be instrumental is as the non-completely simple pseudovarieties
that are closed under two-sided Karnofsky-Rhodes expansion.
