We have monitored the photometric, spectroscopic and polarimetric variations of the classical T Tauri star (CTTS) AA Tau over a period of a month. The light curve consists of more than 260 measurements in each of the B and V-bands over a continuous time period of 30 days and more than 180 measurements in the R and I-bands. This provides unprecedented detail of the photometric variations of a CTTS on timescales ranging from hours to weeks. We find that AA Tau's light curve is quite unlike that of most other CTTS. It exhibits a roughly constant brightness level, interrupted by quasi-cyclic fading episodes with an amplitude of 1.4 mag in BVRI filters. We interpret this behaviour as resulting from quasi-periodic occultations of the stellar photosphere by opaque circumstellar material. The interpretation derives from the lack of significant color variations associated with the fading of the system and is strengthened by the higher polarization level measured when the system is faint. We argue that the occultations are produced by a warp in AA Tau's inner disk which presumably results from the dynamical interaction between the disk and the stellar magnetosphere. We present a model that accounts for the observations quite naturally if we assume that the stellar magnetosphere is a large-scale dipole tilted onto the stellar rotational axis which disrupts the disk at the corotation radius. We derive the geometrical properties of AA Tau's accretion zone in the framework of this model and constrain the location of veiling and Balmer line emitting regions. Although AA Tau's light curve is atypical, the constraints derived here on the structure of its accretion zone may apply as well to other CTTS. It is probably only because AA Tau is seen at a peculiar inclination, close to edge-on, that occultations are conspicuous and its photometric behaviour so clearly reveals this phenomenon. Based on observations obtained at Haute-Provence and Pic du Midi Observatories (France), San Pedro Mártir and Cananea Observatories (México), ESO La Silla Observatory (Chile), Steward Observatory (USA) and Teide Observatory (Spain)}