The hybrid metal−organic interaction that molecular monolayers experience when deposited on metallic substrates can compete with intermolecular interactions and produce a variety of on-surface orders. The variety is already important when the molecule/substrate interface remains atomically flat. Here, we address the case of fullerene molecules deposited at room temperature on a Cu(111) surface with interface atomic-scale roughening. Surprisingly, 10 distinct copper−fullerene phases are found to coexist. Their structure, i.e., the surface relative fullerene/Cu composition and commensurability relationship with the substrate, is resolved using scanning tunneling microscopy and density functional theory calculations. This diversity in the surface roughening process is associated with the multiple possibilities a fullerene can bind to the Cu surface. The molecules are indeed found to have in-plane and out-of-plane positional degrees of freedom: the molecular phases have elastic in-plane properties and can buckle. In addition, the molecules can rotate on their binding sites, conferring extra degrees of freedom to the system. We introduce a competing interaction energy model, parametrized against the results of ab initio calculations, that describes well all the phases we observe experimentally.