  We show that crossing symmetry of S-matrices is modified in certain theories
with non-invertible symmetries or anomalies. Focusing on integrable flows to
gapped phases in two dimensions, we find that S-matrices derived previously
from the bootstrap approach are incompatible with non-invertible symmetries
along the flow. We present consistent alternatives, which however violate
standard crossing symmetry and obey modified rules dictated by fusion
categories. We extend these rules to theories with discrete anomalies.
