We study mode interaction mechanisms in a plasmonic microcavity which consists of two periodically corrugated metallic mirrors. In this system it can be observed that not only does mode hybridization between surface plasmons on the two mirrors occur, but also an anticrossing due to the interaction between plasmons and the Fabry-Perot cavity modes. This behavior is interpreted from the viewpoint of mode symmetry. By controlling the mode interaction strength via structural parameters of the resonant mirrors, the mode dispersion of both the Fabry-Perot cavity mode and the surface plasmons can be modified. This can be applied to control light-matter interaction, subwavelength imaging, or other functional devices.