Even though the physical nature of shear and longitudinal moduli are different, empirical correlations between them have been reported in several biological systems. This correlation is of fundamental interest and immense practical value in biomedicine due to the importance of the shear modulus and the possibility to map the longitudinal modulus at high-resolution with all-optical spectroscopy. We investigate the origin of such a correlation in hydrogels. We hypothesize that both moduli are influenced in the same direction by underlying physicochemical properties, which leads to the observed material-dependent correlation. Matching theoretical models with experimental data, we quantify the scenarios in which the correlation holds. For polymerized hydrogels, a correlation was found across different hydrogels through a common dependence on the effective polymer volume fraction. For hydrogels swollen to equilibrium, the correlation is valid only within a given hydrogel system, as the moduli are found to have different scalings on the swelling ratio. The observed correlation allows one to extract one modulus from another in relevant scenarios.