Diamondoids are a unique form of carbon nanostructure best described as hydrogen-terminated diamond molecules. Their diamond-cage structures and tetrahedral sp3 hybrid bonding create new possibilities for tuning electronic bandgaps, optical properties, thermal transport and mechanical strength at the nanoscale. The recently discovered higher diamondoids have thus generated much excitement in regards to their potential versatility as nanoscale devices. Despite this excitement, however, very little is known about the properties of isolated diamondoids on metal surfaces, a very relevant system for molecular electronics. For example, it is unclear how the microscopic characteristics of molecular orbitals and local electron-vibrational coupling affect electron conduction, emission and energy transfer in the diamondoids. Here, we report the first single-molecule study of tetramantane diamondoids on Au(111) using scanning tunnelling microscopy and spectroscopy. We find that the diamondoid electronic structure and electron-vibrational coupling exhibit unique and unexpected spatial correlations characterized by pronounced nodal structure across the molecular surfaces. Ab initio pseudopotential density functional calculations reveal that much of the observed electronic and vibronic properties of diamondoids are determined by surface hydrogen terminations, a feature having important implications for designing future diamondoid-based molecular devices.