  We give a criterion based on reflection symmetries in the spirit of
Jitomirskaya--Simon to show absence of point spectrum for (split-step) quantum
walks and Cantero--Moral--Vel\'azquez (CMV) matrices. To accomplish this, we
use some ideas from a recent paper by the authors and their collaborators to
implement suitable reflection symmetries for such operators. We give several
applications. For instance, we deduce arithmetic delocalization in the phase
for the unitary almost-Mathieu operator and singular continuous spectrum for
generic CMV matrices generated by the Thue--Morse subshift.
