Polymer chains are attached to nanoparticle surfaces for many purposes, including altering solubility, influencing aggregation, dispersion, and even tailoring immune responses in drug delivery. The most unique structural motif of polymer-grafted nanoparticles (PGNs) is the high-density region in the corona where polymer chains are stretched under significant confinement, but orientation of these chains has never been measured because conventional nanoscale-resolved measurements lack sensitivity to polymer orientation in amorphous regions. Here, we directly measure local chain orientation in polystyrene grafted gold nanoparticles using polarized resonant soft X-ray scattering (P-RSoXS). Using a computational scattering pattern simulation approach, we measure the thickness of the anisotropic region of the corona and extent of chain orientation within it. These results demonstrate the power of P-RSoXS to discover and quantify orientational aspects of structure in amorphous soft materials and provide a framework for applying this emerging technique to more complex, chemically heterogeneous systems in the future.