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Abstract Using integrability, we construct (to leading order in perturbation theory) the explicit form of twist-three light-ray operators in planar N $$ \mathcal{N} $$ = 4 SYM. This construction allows us to directly compute analytically continued CFT data at complex spin. We derive analytically the “magic” decoupling zeroes previously observed numerically. Using the Baxter equation, we also show that certain Regge trajectories merge together into a single unifying Riemann surface. Perhaps more surprisingly, we find that this unification of Regge trajectories is not unique. If we organize twist-three operators differently into what we call “cousin trajectories” we find infinitely more possible continuations. We speculate about which of these remarkable features of twist-three operators might generalize to other operators, other regimes and other theories.