  The goal of the paper is to study the structure of the k-tuples of doubly
$\Lambda$-commuting row isometries and the $C^*$-algebras they generate from
the point of view of noncommutative multivariable operator theory. We obtain
Wold decompositions, in this setting, and use them to classify the $k$-tuples
of doubly $\Lambda$-commuting row isometries up to a unitary equivalence. We
introduce a universal model in this setting, describe its invariant subspaces,
and develop a dilation theory on $\Lambda$-polyballs.
