Short distance scaling limits of a class of integrable models on two-dimensional Minkowski space are considered in the algebraic framework of quantum field theory. Making use of the wedge-local quantum fields generating these models, it is shown that massless scaling limit theories exist, and decompose into (twisted) tensor products of chiral, translation-dilation covariant field theories. On the subspace which is generated from the vacuum by the observables localized in finite light ray intervals, this symmetry can be extended to the M\"obius group. The structure of the interval-localized algebras in the chiral models is discussed in two explicit examples.