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We investigate, in a four photon interference experiment in a laser-writtenwaveguide structure, how symmetries control the suppression of many-body outputevents of a $J_x$ unitary. We show that totally destructive interference doesnot require mutual indistinguishability between all, but only betweensymmetrically paired particles, in agreement with recent theoreticalpredictions. The outcome of the experiment is well described by a quantitativesimulation which accounts for higher order emission of the photon source,imbalances in the scattering network, partial distinguishability, and photonloss.

Symmetry allows for distinguishability in totally destructive many-particle interference