  We quantify our current knowledge of the size and flavor structure of the
matter effects in the evolution of atmospheric and long-baseline neutrinos
based solely on the analysis of the corresponding neutrino data. To this aim we
generalize the matter potential of the Standard Model by rescaling its
strength, rotating it away from the e-e sector, and rephasing it with respect
to the vacuum term. This phenomenological parametrization can be easily
translated in terms of non-standard neutrino interactions in matter. We show
that in the most general case, the strength of the potential cannot be
determined solely by atmospheric and long-baseline data. However its flavor
composition is very much constrained and the present determination of the
neutrino masses and mixing is robust under its presence. We also present an
update of the constraints arising from this analysis in the particular case in
which no potential is present in the e-mu and e-tau sectors. Finally we
quantify to what degree in this scenario it is possible to alleviate the
tension between the oscillation results for neutrinos and antineutrinos in the
MINOS experiment and show the relevance of the high energy part of the spectrum
measured at MINOS.
