  Testing, in a non-trivial, model-independent way, the hypothesis that the
three-massive-neutrinos paradigm properly describes nature is among the main
goals of the current and the next generation of neutrino oscillation
experiments. In the coming decade, the DUNE and Hyper-Kamiokande experiments
will be able to study the oscillation of both neutrinos and antineutrinos with
unprecedented precision. We explore the ability of these experiments, and
combinations of them, to determine whether the parameters that govern these
oscillations are the same for neutrinos and antineutrinos, as prescribed by the
CPT-theorem. We find that both DUNE and Hyper-Kamiokande will be sensitive to
unexplored levels of leptonic CPT-violation. Assuming the parameters for
neutrino and antineutrinos are unrelated, we discuss the ability of these
experiments to determine the neutrino and antineutrino mass-hierarchies,
atmospheric-mixing octants, and CP-odd phases, three key milestones of the
experimental neutrino physics program. Additionally, if the CPT-theorem is
violated in nature in a way that is consistent with all present neutrino and
antineutrino oscillation data, we find that DUNE and Hyper-Kamiokande have the
potential to ultimately establish CPT-invariance violation.
