  The interface between the insulators LaAlO$_3$ and SrTiO$_3$ accommodates a
two-dimensional electron liquid (2DEL) -- a high mobility electron system
exhibiting superconductivity as well as indications of magnetism and
correlations. While this flagship oxide heterostructure shows promise for
electronics applications, the origin and microscopic properties of the 2DEL
remain unclear. The uncertainty remains in part because the electronic
structures of such nanoscale buried interfaces are difficult to probe, and is
compounded by the variable presence of oxygen vacancies and coexistence of both
localized and delocalized charges. These various complications have precluded
decisive tests of intrinsic electronic and orbital reconstruction at this
interface. Here we overcome prior difficulties by developing an interface
analysis based on the inherently interface-sensitive resonant x-ray
reflectometry. We discover a high charge density of 0.5 electrons per
interfacial unit cell for samples above the critical LaAlO$_3$ thickness, and
extract the depth dependence of both the orbital and electronic reconstructions
near the buried interface. We find that the majority of the reconstruction
phenomena are confined to within 2 unit cells of the interface, and we quantify
how oxygen vacancies significantly affect the electronic system. Our results
provide strong support for the existence of polarity induced electronic
reconstruction, clearly separating its effects from those of oxygen vacancies.
