AA Tau is the archetype for a class of stars with a peculiar periodic photometric variability thought to be related to a warped inner disk structure with a nearly edge-on viewing geometry. We present high resolution (∼0 2) ALMA observations of the 0.87 and 1.3 mm dust continuum emission from the disk around AA Tau. These data reveal an evenly spaced three-ringed emission structure, with distinct peaks at 0 34, 0 66, and 0 99, all viewed at a modest inclination of 59 1 0 3 (decidedly not edge-on). In addition to this ringed substructure, we find non-axisymmetric features, including a "bridge" of emission that connects opposite sides of the innermost ring. We speculate on the nature of this "bridge" in light of accompanying observations of HCO+ and 13CO (J = 3-2) line emission. The HCO+ emission is bright interior to the innermost dust ring, with a projected velocity field that appears rotated with respect to the resolved disk geometry, indicating the presence of a warp or inward radial flow. We suggest that the continuum bridge and HCO+ line kinematics could originate from gap-crossing accretion streams, which may be responsible for the long-duration dimming of optical light from AA Tau.