  We propose a natural $\mathbb{Z}_2 \times \mathbb{Z}_2$-graded generalisation
of $d=2$, $\mathcal{N}=(1,1)$ supersymmetry and construct a
$\mathbb{Z}_2^2$-space realisation thereof. Due to the grading, the
supercharges close with respect to, in the classical language, a commutator
rather than an anticommutator. This is then used to build classical (linear and
non-linear) sigma models that exhibit this novel supersymmetry via mimicking
standard superspace methods. The fields in our models are bosons, right-handed
and left-handed Majorana-Weyl spinors, and exotic bosons. The bosons commute
with all the fields, the spinors belong to different sectors that cross commute
rather 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.
