We investigate fragmentation processes induced by core-level photoexcitation in CF4 at both the carbon and fluorine K edges by means of partial-ion-yield spectroscopy. The molecule CF4 is a textbook example of systems in which Jahn-Teller coupling strongly manifests itself in the photoabsorption spectrum. Spectral features related to Jahn-Teller and quasi-Jahn-Teller splitting are observed, and important differences in the fragmentation pathways are revealed depending on the symmetries of the core-excited states. We interpret these experimental observations on the grounds of symmetry lowering from the Td to the C3v point group as well as preferential orientation with respect to the polarization vector of the incident radiation