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We describe two new discrete symmetries of the inviscid Burgers (or Riemann-Hopf) equation ut+uux=0. We derived both of them using a local, formal approach of Hopf algebraic renormalization, a tool recently used in algorithmic computations. We prove that one of them is a Lie point transformation. Symmetries generate new exact solutions from the known solutions and provide useful frames of reference in the study of shock wave formation.