Both experiments and theoretical studies have demonstrated that the interaction between the current-carrying electrons and the induced polarization charge in single-molecule junctions leads to a strong renormalization of molecular charging energies. However, the effect on electronic excitations and molecular symmetries remain unclear. Using a theoretical framework developed for semiconductor-nanostructure-based single-electron transistors (SETs), we demonstrate that the image charge interaction breaks the molecular symmetries in a benzene-based single-molecule transistor operating in the Coulomb blockade regime. This results in the appearance of a so-called blocking state, which gives rise to negative-differential resistance (NDR). We show that the appearance of NDR and its magnitude in the symmetry-broken benzene SET depends in a complicated way on the interplay between the many-body matrix elements, the lead tunnel coupling asymmetry, and the bias polarity. In particular, the current reducing property of the blocking state causing the NDR is shown to vanish under strongly asymmetric tunnel couplings, when the molecule is coupled stronger to the drain electrode. The calculated I-V characteristic may serve as an indicator for image charge broken molecular symmetries in experimental situations.