A detailed comparison of tartaric acid (HOOC-CHOH-CHOH-COOH) and succinic acid (HOOC-CH2-CH2-COOH) molecules on a Cu(110) surface is presented with a view to elucidate how the two-dimensional chirality exhibited by such robust, chemisorbed systems is affected when both OH groups of the former molecule are replaced with H groups, a stereochemical change that leaves the metal-bonding functionalities of the molecule untouched but destroys both chiral centers. It is found that this change does not significantly affect the thermodynamically preferred chemical forms that are adopted, namely the doubly deprotonated bicarboxylate at low coverages (theta less than or equal to (1)/(6) ML) and the singly deprotonated monocarboxylate at higher coverage. However, the kinetics of phase formation are significantly affected so that the conditions required for self-assembling pertinent two-dimensional chiral phases alter substantially. For both molecules, two-dimensional assembly is found to depend strongly on the nature of the local adsorption motif created, with each motif essentially acting as a "synthon" for the supramolecular assembly. In this respect, it seems that molecule-metal bonding interactions define the general self-assembly structure. The presence/absence of the OH groups, instead, cause a subtler, second-order effect on the finer details of the self-assembled structure. Finally, the creation of chirality in the achiral succinate system is shown to arise from adsorption-induced asymmetrization, inducing point chirality via molecular distortion and/or metal reconstruction of the local adsorption unit. This chiral adsorption unit is then responsible for creating chiral supramolecular through-space and through-metal interactions that propagate a chiral organization. However, the achirality of the succinate ensures that nucleation points of either chirality are equally created, producing a racemic conglomerate of coexisting mirror domains. It is in this aspect that the uniquely aligned OH groups of the rigid bitartrate system wield the greatest effect, by favoring one distortion/reconstruction for the (R, R)-bitartrate and its mirror image distortion/reconstruction for the (S,S)-enantiomer, creating surfaces that are globally chiral on the macroscopic scale. So overall, the OH groups do not dictate the general nature of the assembly but are critical as chiral propagators, breaking the degeneracy and thus promoting asymmetry to chirality.