This paper reports measurements of atmospheric neutrino and antineutrino interactions in the MINOS Far Detector, based on 2553 live-days (37.9 kton-years) of data. A total of 2072 candidate events are observed. These are separated into 905 contained-vertex muons and 466 neutrino-induced rock-muons, both produced by charged-current $ u_{mu}$ and $bar{ u}_{mu}$ interactions, and 701 contained-vertex showers, composed mainly of charged-current $ u_{e}$ and $bar{ u}_{e}$ interactions and neutral-current interactions. The curvature of muon tracks in the magnetic field of the MINOS Far Detector is used to select separate samples of $ u_{mu}$ and $bar{ u}_{mu}$ events. The observed ratio of $bar{ u}_{mu}$ to $ u_{mu}$ events is compared with the Monte Carlo simulation, giving a double ratio of $R^{data}_{bar{ u}/ u}/R^{MC}_{bar{ u}/ u} = 1.03 pm 0.08 (stat.) pm 0.08 (syst.)$. The $ u_{mu}$ and $bar{ u}_{mu}$ data are separated into bins of $L/E$ resolution, based on the reconstructed energy and direction of each event, and a maximum likelihood fit to the observed $L/E$ distributions is used to determine the atmospheric neutrino oscillation parameters. This fit returns 90% confidence limits of $|Delta m^{2}| = (1.9 pm 0.4) times 10^{-3} eV^{2}$ and $sin^{2} 2theta > 0.86$. The fit is extended to incorporate separate $ u_{mu}$ and $bar{ u}_{mu}$ oscillation parameters, returning 90% confidence limits of $|Delta m^{2}|-|Delta bar{m}^{2}| = 0.6^{+2.4}_{-0.8} times 10^{-3} eV^{2}$ on the difference between the squared-mass splittings for neutrinos and antineutrinos.
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