Riemann’s method is one of the definitive ways of solving Cauchy’s problem for a second order linear hyperbolic partial differential equation in 2 variables. Chaundy’s equation, with 4 parameters, is the most general self-adjoint equation for which the Riemann function is known. Here we show that Chaundy’s equation possesses a two-dimensional vector space of second-order symmetry operators. Hence a new equivalence class of Riemann functions, admitting no first-order symmetries and obtainable only via a higher order symmetry, is found. A new 5 parameter Riemann function is then subsequently derived.