  We show that non-standard neutrino interactions (NSI) can notably modify the
pattern of resonant flavor conversion of neutrinos within supernovae and
significantly impact the neutronization burst signal in forthcoming experiments
such as the Deep Underground Neutrino Experiment (DUNE). The presence of NSI
can invert the energy levels of neutrino matter eigenstates and even induce a
new resonance in the inner parts close to the proto-neutron star. We
demonstrate how DUNE can use these new configurations of energy levels to have
sensitivity to NSIs down to $\mathcal{O}(0.1)$. We also elucidate how the
effect may result in a puzzling confusion of normal and inverted mass orderings
by highlighting the emergence or vanishing of the neutronization peak, which
distinguishes between the two mass orderings. Potential implications are
analyzed thoroughly.
