We study the problem of relating cycles on a triod Y to circle rotations. We prove that a triod-twist cycle P, the simplest cycle on a triod with a given rotation number ρ, is conjugate to circle rotation, by angle ρ, restricted to one of its cycles Q. Further, the conjugacy Ψ : P → Q is piece-wise monotone and its modality exceeds the modality m of P by at-most 3. This explicit bound m+3 serves as a combinatorial distortion principle, where the additive constant “+3” represents the “topological cost” imposed by the valence of the branching point a.