© 2016 American Physical Society. We investigate bond-rearrangement driven by photo-double-ionization (PDI) near and above the double-ionization threshold in a sequence of carbon-carbon double-bonded hydrocarbon molecules: ethylene, fluoroethylene, and 1,1-difluoroethylene. We employ the kinematically complete cold target recoil ion momentum spectroscopy method to resolve all photo-double-ionization events leading to two-ion fragments. We observe changes in the branching ratios of different dissociative ionization channels depending on the presence of no, one, or two fluorine atoms. The role of the fluorine atom in the bond-rearrangement channels is intriguing, as evident by the reordering of the threshold energies of the PDI in the fluorinated molecules. These effects offer a compelling argument that the electronegativity of the fluorine (or the polarity of the molecule) strongly influences the potential energy surfaces of the molecules and drives bond rearrangement during the dissociation process. The energy sharing and the relative angle between the three-dimensional momentum vectors of the two electrons enable us to distinguish between knockout and other ionization mechanisms of the PDI processes.