High-resolution neutron diffraction has been used in conjunction with hydrogen/deuterium isotopic labeling to determine with unprecedented detail the structure of two archetypal aromatic liquids: benzene and toluene. We discover the nature of aromatic pi-pi interactions in the liquid state by constructing for the first time a full six-dimensional spatial and orientational picture of these systems. We find that in each case the nearest neighbor coordination shell contains approximately 12 molecules. Benzene is the more structured of the two liquids, showing, for example, a sharper nearest neighbor coordination peak in the radial distribution function. Superficially the first neighbor shells appear isotropic, but our multidimensional analysis shows that the local orientational order in these liquids is much more complex. At small molecular separations (<5 angstrom) there is a preference for parallel pi-pi contacts in which the molecules are offset to mimic the interlayer structure of graphite. At larger separations (>5 angstrom) the neighboring aromatic rings are predominantly perpendicular, with two H atoms per molecule directed toward the acceptor's pi orbitals. The so-called "anti-hydrogen-bond" configuration, proposed as the global minimum for the benzene dimer, occurs only as a saddle point in our data. The observed liquid structures are therefore fundamentally different than those deduced from the molecular dimer energy surfaces.