  We investigate, in a four photon interference experiment in a laser-written
waveguide structure, how symmetries control the suppression of many-body output
events of a $J_x$ unitary. We show that totally destructive interference does
not require mutual indistinguishability between all, but only between
symmetrically paired particles, in agreement with recent theoretical
predictions. The outcome of the experiment is well described by a quantitative
simulation which accounts for higher order emission of the photon source,
imbalances in the scattering network, partial distinguishability, and photon
loss.
