We present integral field spectroscopy of two selected zones in the Orion nebula obtained with the Potsdam Multi-Aperture Spectrophotometer, covering the optical spectral range from 3500 to 7200 Å and with a spatial resolution of 1 arcsec. The observed zones are located on the prominent Bright bar and on the brightest area at the north-east of the Orion south cloud, both containing remarkable ionization fronts. We obtain maps of emission-line fluxes and ratios, electron density and temperatures, and chemical abundances. We study the ionization structure and morphology of both fields, whose ionization fronts show different inclination angles with respect to the plane of the sky. We find that the maps of electron density, O+/H+ and O/H ratios show a rather similar structure. We interpret this as produced by the strong dependence on density of the [O II] lines used to derive the O+ abundance, and that our nominal values of electron density - derived from the [S II] line ratio - may be slightly higher than the appropriate value for the O+ zone. We measure the faint recombination lines of O II in the field at the north-east of the Orion south cloud, allowing us to explore the so-called abundance discrepancy problem. We find a rather constant abundance discrepancy across the field and a mean value similar to that determined in other areas of the Orion nebula, indicating that the particular physical conditions of this ionization front do not contribute to this discrepancy. Based on observations collected at the Centro Astronómico Hispano Alemán (CAHA) at Calar Alto, operated jointly by the Max-Planck Institut für Astronomie and the Instituto de Astrofísica de Andalucía (CSIC).