Covalent functionalizations represent a very promising avenue to engineer or manipulate carbon nanotubes. However, in metallic tubes the electrical conductance can drop by several orders of magnitude following functionalization, due to sp(3) rehybridization of the sidewall carbons that strongly disrupts the conjugated pi-network. First-principles calculations have predicted that some divalent functional groups, carbenes or nitrenes, can instead recover the original sp(2) hybridization and perfect metallic conductance of the pristine tubes. In these cycloaddition reactions, the extra bond added by the functional group with each of the bridgehead carbons is compensated by a breaking of the sidewall bond between them, restoring in the process the original sp(2) environment. We characterize this bond-breaking chemistry with extensive first-principles calculations and highlight its sensitivity to the orientation of the pi-electron system of the chosen addend. Using dinitrocarbene as a model case, we show that the bridgehead carbon atoms can reversibly rehybridize from sp(2) to sp(3) in response to the pi orientation of the addends. These results suggest a novel route to modulating the electronic properties of carbon nanotubes that is based on orbital rehybridization and that can be directed with optical or electrochemical means.