  One of the main goals of the Deep Underground Neutrino Experiment (DUNE) is
to look for new sources of CP-invariance violation. Another is to significantly
test the three-massive-neutrinos paradigm. Here, we show that there are
CP-invariant new physics scenarios which, as far as DUNE data are concerned,
cannot be distinguished from the three-massive-neutrinos paradigm with very
large CP-invariance violating effects. We discuss examples with non-standard
neutrino interactions and with a fourth neutrino mass eigenstate. We briefly
discuss how ambiguities can be resolved by combining DUNE data with data from
other long-baseline experiments, including Hyper-Kamiokande.
