  We introduce a subclass of the commutative regular languages that is
characterized by the property that the state set of the minimal deterministic
automaton can be written as a certain Cartesian product. This class behaves
much better with respect to the state complexity of the shuffle, for which we
find the bound~$2nm$ if the input languages have state complexities $n$ and
$m$, and the upward and downward closure and interior operations, for which we
find the bound~$n$. In general, only the bounds $(2nm)^{|\Sigma|}$ and
$n^{|\Sigma|}$ are known for these operations in the commutative case. We prove
different characterizations of this class and present results to construct
languages from this class. Lastly, in a slightly more general setting of
partial commutativity, we introduce other, related, language classes and
investigate the relations between them.
