[en] Free string theory on the plane-wave background displays a discrete Z₂ symmetry exchanging the two transverse SO(4) groups. This symmetry should be respected also at the interacting level. We show that the zero mode structure proposed in hep-th/0208148 can be completed to a full kinematical vertex, contrary to claims appeared in the previous literature. We also comment on the relation with recent works on the string-bit formalism and on the comparison with the field theory side of the correspondence. (Abstract Copyright [2003], Wiley Periodicals, Inc.)