Buoyancy-restored modes inside neutron stars depend sensitively on both the microphysical (e.g., composition and entropy gradients) and macrophysical (e.g., stellar mass and radius) properties of the star. Asteroseismology efforts for $g$-modes are therefore particularly promising avenues for recovering information concerning the nuclear equation of state. In this work it is shown that the overall low-temperature $g$-space consists of multiple groups corresponding to different classes of equation of state (e.g., hadronic vs. hybrid). This is in contrast to the case of pressure-driven modes, for example, which tend to follow a universal relation regardless of microphysical considerations. Using a wide library of currently-viable equations of state, perturbations of static, stratified stars are calculated in general relativity to demonstrate in particular how $g$-space groupings can be classified according to the mean mass density, temperature, central speed of sound, and tidal deformability. Considering present and future observations regarding gravitational waves, accretion outbursts, quasi-periodic oscillations, and precursor flashes from gamma-ray bursts, it is shown how one might determine which group the $g$-modes belong to.