  We discuss relation between the cluster integrable systems and spin chains in
the context of their correspondence with 5d supersymmetric gauge theories. It
is shown that $\mathfrak{gl}_N$ XXZ-type spin chain on $M$ sites is isomorphic
to a cluster integrable system with $N \times M$ rectangular Newton polygon and
$N \times M$ fundamental domain of a 'fence net' bipartite graph. The Casimir
functions of the Poisson bracket, labeled by the zig-zag paths on the graph,
correspond to the inhomogeneities, on-site Casimirs and twists of the chain,
supplemented by total spin. The symmetricity of cluster formulation implies
natural spectral duality, relating $\mathfrak{gl}_N$-chain on $M$ sites with
the $\mathfrak{gl}_M$-chain on $N$ sites. For these systems we construct
explicitly a subgroup of the cluster mapping class group
$\mathcal{G}_\mathcal{Q}$ and show that it acts by permutations of zig-zags
and, as a consequence, by permutations of twists and inhomogeneities. Finally,
we derive Hirota bilinear equations, describing dynamics of the tau-functions
or A-cluster variables under the action of some generators of
$\mathcal{G}_\mathcal{Q}$.
