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 νμ and ν̅ μ interactions, and 701 contained-vertex showers, composed mainly of charged-current νe and ν̅ 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 νμ and ν̅ μ events. The observed ratio of ν̅ μ to νμ events is compared with the Monte Carlo (MC) simulation, giving a double ratio of Rν̅ /νdata/Rν̅ /νMC=1.03±0.08(stat)±0.08(syst). The νμ and ν̅ μ 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 |Δm2|=(1.9±0.4)×10-3 eV2 and sin22θ>0.86. The fit is extended to incorporate separate νμ and ν̅ μ oscillation parameters, returning 90% confidence limits of |Δm2|-|Δm̅ 2|=0.6-0.8+2.4×10-3 eV2 on the difference between the squared-mass splittings for neutrinos and antineutrinos.