Saturn's magnetospheric magnetic field, planetary radio emissions, plasma populations, and magnetospheric structure are all known to be modulated at periods close to the assumed rotation period of the planetary interior. These oscillations are readily apparent despite the high degree of axisymmetry in the internally produced magnetic field of the planet and have different rotation periods in the northern and southern hemispheres. In this paper we study the spatial structure of (near-)planetary period magnetic field oscillations in Saturn's equatorial magnetosphere. Extending previous analyses of these phenomena, we include all suitable data from the entire Cassini mission during its orbital tour of the planet so as to be able to quantify both the amplitude and phase of these field oscillations throughout Saturn's equatorial plane, to distances of 30 planetary radii. We study the structure of these field oscillations in view of both independently rotating northern and southern systems, finding spatial variations in both magnetic fields and inferred currents flowing north-south that are common to both systems. With the greatly expanded coverage of the equatorial plane achieved during the latter years of the mission, we are able to present a complete survey of dawn-dusk and day-night asymmetries in the structure of the oscillating fields and currents. We show that the general structure of the rotating currents is simpler than previously reported and that the relatively enhanced nightside equatorial fields and currents are due in part to related periodic vertical motion of Saturn's magnetotail current sheet