Metal-organic interfaces are critical in organic electronic devices, influencing key performance properties. Understanding these relationships is essential for improving such devices. Polycyclic conjugated hydrocarbons (PCHs) with alternant and non-alternant topologies are promising candidates for exploring these interfaces since they show physisorption and chemisorption, respectively. Using density functional theory with periodic boundary conditions, we modeled the interfaces between a Cu(111) surface and 22 PCHs (11 alternant and 11 non-alternant). We identified quantitative correlations among interface properties, showing that these properties form a "fixed set" of properties for individual molecules. A clear distinction emerges between physisorption and chemisorption for most properties, except for work function changes, which are consistently governed by the Pauli pushback effect resulting from dispersion pull. Interestingly, molecules with larger π-electron systems exhibit stronger dispersion attraction yet higher adsorption heights. This study provides chemically intuitive explanations for these findings and highlights the interconnected nature of interface properties. The insights gained offer valuable guidance for understanding and optimizing Cu(111)-organic interfaces, contributing to advancements in organic electronics.