  This paper describes a generalization of decomposition in orbifolds. In
general terms, decomposition states that two-dimensional orbifolds and gauge
theories whose gauge groups have trivially-acting subgroups decompose into
disjoint unions of theories. However, decomposition can be, at least naively,
broken in orbifolds if the orbifold has discrete torsion in the
trivially-acting subgroup. (Formally, this breaks finite global one-form
symmetries.) Nevertheless, even in such cases, one still sees rudiments of
decomposition. In this paper, we generalize decomposition in orbifolds to
include such examples of discrete torsion, which we check in numerous examples.
Our analysis includes as special cases (and in one sense generalizes) quantum
symmetries of abelian orbifolds.
