Structural, electronic, and vibrational properties of the endohedral fullerene Ce@C82 on Cu(111) have been studied by scanning tunneling microscopy (STM) and density functional theory (DFT). Ce@C82 forms islands on the substrate. Our STM measurements show relatively large differences in morphology and electron spectra of molecules within these islands indicating multiple molecular orientations on the substrate, while the vibrational spectra are more uniform. We have determined molecular orientations by comparing STM and DFT molecular morphology, and we have calculated Ce@C82 bound to Cu(111) and found that it is chemisorbed. We show that Ce@C82 adopts orientations on the surface that enables Ce to remain at its most favorable binding site inside C82. The effect of chemisorption on the structural and electronic properties of Ce@C82 is thus small, and the orientations are limited to configurations with Ce in the upper hemisphere of the molecular configurational space. We show that the variations in the dI/dV spectra between molecules of different orientations is due to Ce-cage orbitals that are localized in space and their involvement in tunneling depends on the molecular orientation on the substrate. The observed electron-vibration coupling modes in the STM-IETS (in-elastic tunneling spectroscopy) of Ce@C82 arise from cage modes only, and therefore, electron transport properties are expected to be different compared to Ce2@C80, which has active Ce-cage vibrations.