Our previous constructions of Borchers triples are extended to massless scattering with nontrivial left and right components. A massless Borchers triple is constructed from a set of left-left, right-right and left-right scattering functions. We find a correspondence between massless left-right scattering S-matrices and massive block diagonal S-matrices. We point out a simple class of S-matrices with examples. We study also the restriction of two-dimensional models to the lightray. Several arguments for constructing strictly local two-dimensional nets are presented and possible scenarios are discussed.