The large-scale magnetic field of our Galaxy can be probed in three dimensions using Faraday rotation of pulsar signals. We report on the determination of 223 rotation measures from polarization observations of relatively distant southern pulsars made using the Parkes radio telescope. Combined with previously published observations, these data give clear evidence for large-scale counterclockwise fields (viewed from the north Galactic pole) in the spiral arms interior to the Sun and weaker evidence for a counterclockwise field in the Perseus arm. However, in interarm regions, including the solar neighborhood, we present evidence that suggests that large-scale fields are clockwise. We propose that the large-scale Galactic magnetic field has a bisymmetric structure with reversals on the boundaries of the spiral arms. Streaming motions associated with spiral density waves can directly generate such a structure from an initial, inwardly directed radial field. Large-scale fields increase toward the Galactic center, with a mean value of about 2 kG in the solar neighborhood and 4 kG at a galactocentric radius of 3 kpc.