  Two matrix product states (MPS) are in the same phase in the presence of
symmetries if they can be transformed into one another via symmetric
short-depth circuits. We consider how symmetry-preserving measurements with
feedforward alter the phase classification of MPS in the presence of global
on-site symmetries. We demonstrate that, for all finite abelian symmetries, any
two symmetric MPS belong to the same phase. We give an explicit protocol that
achieves a transformation between any two phases and that uses only a depth-two
symmetric circuit, two rounds of symmetric measurements, and a constant number
of auxiliary systems per site. In the case of non-abelian symmetries, symmetry
protection prevents one from deterministically transforming symmetry-protected
topological (SPT) states to product states directly via measurements, thereby
complicating the analysis. Nonetheless, we provide protocols that allow for
asymptotically deterministic transformations between the trivial phase and
certain SPT phases.
