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Quantitative theoretical techniques for understanding the substructure of jets at the LHC enable new insights into the dynamics of QCD, and novel approaches to search for new physics. Recently, there has been a program to reformulate jet substructure in terms of correlation functions, $\langle \mathcal{E}(\vec n_1) \mathcal{E}(\vec n_2) \cdots \mathcal{E}(\vec n_k) \rangle$, of light-ray operators, $\mathcal{E}(\vec n)$, allowing the application of techniques developed in the study of Conformal Field Theories (CFTs). In this paper we further develop these techniques in the particular context of the three-point correlator $\langle \mathcal{E}(\vec n_1) \mathcal{E}(\vec n_2) \mathcal{E}(\vec n_3) \rangle$, using recently computed perturbative data in both QCD and $\mathcal{N}=4$ sYM. We derive the celestial blocks appearing in the light-ray operator product expansion (OPE) of the three-point correlator, and use the Lorentzian inversion formula to extract the spectrum of light-ray operators appearing in the expansion, showing, in particular, that the OPE data is analytic in transverse spin. Throughout our presentation, we highlight the relation between the OPE approach, and more standard splitting function based approaches of perturbative QCD, emphasizing the utility of the OPE approach for incorporating symmetries in jet substructure calculations. We hope that our presentation introduces a number of new techniques to the jet substructure community, and also illustrates the phenomenological relevance of the study of light-ray operators in the OPE limit to the CFT community.