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We propose a natural $\mathbb{Z}_2 \times \mathbb{Z}_2$-graded generalisationof $d=2$, $\mathcal{N}=(1,1)$ supersymmetry and construct a$\mathbb{Z}_2^2$-space realisation thereof. Due to the grading, thesupercharges close with respect to, in the classical language, a commutatorrather than an anticommutator. This is then used to build classical (linear andnon-linear) sigma models that exhibit this novel supersymmetry via mimickingstandard superspace methods. The fields in our models are bosons, right-handedand left-handed Majorana-Weyl spinors, and exotic bosons. The bosons commutewith all the fields, the spinors belong to different sectors that cross commuterather than anticommute, while the exotic boson anticommute with the spinors.As a particular example of one of the models, we present a `double-graded'version of supersymmetric sine-Gordon theory.

$\mathbb{Z}_2\times \mathbb{Z}_2$-graded supersymmetry: $2$-d sigma models