  We examine the question of the integrability of the recently defined
$\mathbb{Z}_2\times \mathbb{Z}_2$-graded sine-Gordon model, which is a natural
generalisation of the supersymmetric sine-Gordon equation. We do this via
appropriate auto-B\"{a}cklund transformations, construction of conserved
spinor-valued currents and a pair of infinite sets of conservation laws.
