Anisotropy of intermolecular and molecule-substrate interactions holds the key to controlling the arrangement of fullerenes into 2D self-assembled monolayers (SAMs). The chemical reactivity of fullerenes allows functionalization of the carbon cages with sulfur-containing groups, thiols and thioethers, which facilitates the reliable adsorption of these molecules on gold substrates. A series of structurally related molecules, eight of which are new fullerene compounds, allows systematic investigation of the structural and functional parameters defining the geometry of fullerene SAMs. Scanning tunnelling microscopy (STM) measurements reveal that the chemical nature of the anchoring group appears to be crucial for the long-range order in fullerenes: the assembly of thiol-functionalized fullerenes is governed by strong molecule-surface interactions, which prohibit formation of ordered molecular arrays, while thioether-functionalized fullerenes, which have a weaker interaction with the surface than the thiols, form a variety of ordered 2D molecular arrays owing to noncovalent intermolecular interactions. A linear row of fullerene molecules is a recurring structural feature of the ordered SAMs, but the relative alignment and the spacing between the fullerene rows is strongly dependent on the size and shape of the spacer group linking the fullerene cage and the anchoring group. Careful control of the chemical functionality on the carbon cages enables positioning of fullerenes into at least four different packing arrangements, none of which have been observed before. Our new strategy for the controlled arrangement of fullerenes on surfaces at the molecular level will advance the development of practical applications for these nanomaterials.