Nonlinear vibrational spectroscopies such as visible-infrared sum-frequency spectroscopy may serve as powerful probes of interfacial structure. Obtaining quantitative orientation information, however, has been limited by the required knowledge of the corresponding molecular-level nonlinear optical properties. We provide a general scheme for calculating the vibrational hyperpolarizability of any infrared- and Raman-active mode, regardless of the molecular symmetry or complexity of the structure. Our method involves all atoms and therefore does not rely on making any local mode approximations. We show how this information is used together with experimental data to arrive at the tilt and twist angles of a surfactant headgroup at the air/water interface. Since our approach is completely general, it may be used for the analysis of any adsorbate at an isotropic interface.