Double-disk resonators, composed of two nearly-identical dielectric disks separated by nanoscale air gap, can be configured to exhibit small modal volumes with high-Q factors. Compared to the two fundamental eigenmodes of a single-disk resonator (TE and TM), those supported by the double-disk microcavity are split into four modes, which can be categorized as symmetric and antisymmetric modes: TEs/TEas and TMs/TMas, depending on the field symmetry. Theoretical descriptions on these eigenmodes are given with regard to the cavity performance metrics such as cavity mode dispersion, Q-factor, and mode index. Experimental verification of these eigenmodes is provided for a 40-nm gap double-disk/air-slot resonator. In addition to these optical mode characterizations, the mechanical eigenmodes of double-disk structures, which can be actuated by the optical gradient forces, are investigated.