  We investigate superfield formulation of the minimal
$\mathbb{Z}_2^2$-supersymmetry. It is shown that the integrability on
$\mathbb{Z}_2^2$-superspace guarantees the invariance of action. Then we
present two superfields which carry distinct irreducible representations of the
$\mathbb{Z}_2^2$-supersymmetry algebra. One of them gives integrable Lagrangian
and the other does not. We also show that integrable superfields with different
$\mathbb{Z}_2^2$-degree also carry irreducible representations and they give
invariant actions. To perform this analysis, the representation theory of the
minimal $\mathbb{Z}_2^2$-supersymmetry algebra is studied in some detail.
