We study the complementarity between accelerator and reactor coherent elastic neutrino-nucleus elastic scattering (CENS) experiments for constraining new physics in the form of non-standard neutrino interactions (NSI). Firstly, considering just data from the recent observation by the COHERENT experiment, we explore interpretive degeneracies that emerge when activating either two or four unknown NSI parameters. Next, we demonstrate that simultaneous treatment of reactor and accelerator experiments, each employing at least two distinct target materials, can break a degeneracy between up and down flavor-diagonal NSI terms that survives analysis of neutrino oscillation experiments. Considering four flavor-diagonal () up and down-type NSI parameters, we find that all terms can be measured with high local precision (to a width as small as 5% in Fermi units) by next-generation experiments, although discrete reflection ambiguities persist.

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