  Trivially-acting symmetries in two-dimensional conformal field theory include
twist fields of dimension zero which are local topological operators. We
investigate the consequences of regarding these operators as part of the global
symmetry of the theory. That is, we regard such a symmetry as a mix of
topological defect lines (TDLs) and topological point operators (TPOs). TDLs
related by a trivially-acting symmetry can join at a TPO to form non-trivial
two-way junctions. Upon gauging, the local operators at those junctions can
become vacua in a disjoint union of theories. Examining the behavior of the
TPOs under gauging therefore allows us to refine decomposition by tracking the
trivially-acting symmetries of each universe. Mixed anomalies between the TDLs
and TPOs provide discrete torsion-like phases for the partition functions of
these orbifolds, modifying the resulting decomposition. This framework also
readily allows for the consideration of trivially-acting non-invertible
symmetries.
