The conformations of organometallic polymers formed via the bottom-up assembly of monomer units on a metal surface are investigated, and the relationship between the adsorption geometry of the individual monomer units, the conformational structure of the chain, and the overall shape of the polymer is explored. Iodine-functionalized monomer units deposited on a Au(111) substrate are found to form linear chain structures in which each monomer is linked to its neighbors via a Au adatom. Lateral manipulation of the linear chains using a scanning tunneling microscope allows the structure of the chain to be converted from a linear to a curved geometry, and it is shown that a transformation of the overall shape of the chain is coupled to a conformational rearrangement of the chain structure as well as a change in the adsorption geometry of the monomer units within the chain. The observed conformational structure of the curved chain is well-ordered and distinct from that of the linear chains. The structures of both the linear and curved chains are investigated by a combination of scanning tunneling microscopy measurements and theoretical calculations.