  A $\mathbb{Z}_2 \times \mathbb{Z}_2$-graded generalisation of the quantum
superplane is proposed and studied. We construct a bicovariant calculus on what
we shall refer to as the \emph{double-graded quantum superplane}. The
commutation rules between the coordinates, their differentials and partial
derivatives are explicitly given. Furthermore, we show that an extended version
of the double-graded quantum superplane admits a natural Hopf
$\mathbb{Z}_2^2$-algebra structure.
