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In this article, which builds upon the work done in our previous paper, the chiral aspects of 2dcft on globally hyperbolic Lorentzian manifolds are developed and explored within the perturbative algebraic quantum field theory (pAQFT) framework. In the example of the massless scalar field on globally hyperbolic 2-dimensional spacetimes, we identify the subalgebras of a given theory comprising only chiral (or anti-chiral) observables. These subalgebras are constructed explicitly, with the help of structures naturally associated to a Cauchy surface, for both the classical and the quantised theory, and it is shown that they then embed naturally into the algebra of the full theory. Finally, it is demonstrated that the construction of these subalgebras is independent of the choice of Cauchy surface and that they unambiguously define a covariant theory on the spaces of null-geodesics.