  We study the complementarity between accelerator and reactor coherent elastic
neutrino-nucleus elastic scattering (CE$\nu$NS) 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 ($ee/\mu\mu$) up and down-type NSI parameters,
we find that all terms can be measured with high local precision (to a width as
small as $\sim$5\% in Fermi units) by next-generation experiments, although
discrete reflection ambiguities persist.
