We present evidence for soft gluon interference, as required by QCD. This interference is expected to manifest itself in an angular ordering of the gluons radiated within a jet. Using hadronic decays of the Z boson in the L3 detector at LEP, we compare variables sensitive to such an angular ordering, namely the energy-energy correlation asymmetry and the newly introduced particle- particle correlation asymmetry, with the predictions of various parton shower models. Only those models which incorporate the expected interference agree with the data.