Turbulent jets are known to support large-scale vortical wave packets traveling downstream. We show that a propagating helical wave represents a common form of the "optimal" eigenfunction tracking these structures from the near to the far field of a round jet issuing from a pipe. Two first mirror-symmetric modes containing around 5% of the total turbulent kinetic energy capture all significant large-scale events and accurately replicate the full shear-layer dynamics of the azimuthal wave number m=1. A family of the most energy-containing traveling waves represents low wave numbers and is described in terms of "empirical" dispersion laws.