  In this paper, we introduce a new set of modular-invariant phase factors for
orbifolds with trivially-acting subgroups, analogous to discrete torsion and
generalizing quantum symmetries. After describing their basic properties, we
generalize decomposition to include orbifolds with these new phase factors,
making a precise proposal for how such orbifolds are equivalent to disjoint
unions of other orbifolds without trivially-acting subgroups or one-form
symmetries, which we check in numerous examples.
