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We present a solution for the Green's function for the general case of a helical wormlike chain with twist-bend coupling, and demonstrate the applicability of our solution for evaluating general structural and mechanical chain properties. We find that twist-bend coupling renormalizes the persistence length and the force-extension curves relative to worm-like chains. Analysis of intrinsically twisted polymers shows that incorporation of twist-bend coupling results in the oscillatory behavior in principal tangent correlations that are observed in some studies of synthetic polymers. The exact nature of our solution provides a framework to evaluate the role of twist-bend coupling on polymer properties and motivates the reinterpretation of existing bio-polymer experimental data.