  We prove that matrix-product unitaries (MPUs) with on-site unitary symmetries
are completely classified by the (chiral) index and the cohomology class of the
symmetry group $G$, provided that we can add trivial and symmetric ancillas
with arbitrary on-site representations of $G$. If the representations in both
system and ancillas are fixed to be the same, we can define symmetry-protected
indices (SPIs) which quantify the imbalance in the transport associated to each
group element and greatly refines the classification. These SPIs are stable
against disorder and measurable in interferometric experiments. Our results
lead to a systematic construction of two-dimensional Floquet symmetry-protected
topological (SPT) phases beyond the standard classification, and thus shed new
light on understanding nonequilibrium phases of quantum matter.
