The isometries of an exact plane gravitational wave are symmetries for both massive and massless particles. Their conformal extensions are, in fact, chrono-projective transformations {introduced earlier by Duval et al. [Classical Quantum Gravity 3, 461 (1986); Classical Quantum Gravity 32(17), 175006 (2015)]} and are symmetries for massless particles. Homotheties are universal chrono-projective symmetries for any profile. Chrono-projective transformations also generate new conserved quantities for the underlying non-relativistic systems in the Bargmann framework. Homotheties play a similar role for the lightlike “vertical” coordinate as isometries play for the transverse coordinates.