This work reports on the plasmonic properties of a symmetry-breaking system consisting of rhodium dual broken nanorings, in the ultraviolet-visible regime. In the structure, two rhodium broken rings are located with a separation on the scale of nanometers. As the separation, the light polarization, and the relative orientation of the broken angles are respectively varied, the plasmonic scattering efficiency of the system is investigated, using the finite difference time domain method. Multiple plasmonic resonances are revealed, and the associated asymmetry-induced Fano-like lineshapes are fitted to a model that employs multiple Fano lineshape functions. The resonance wavelengths, the spectral widths, and the characteristic q values are determined from the best fit parameters, and the plasmonic characteristics of the system are quantitatively probed. The results in this work may be beneficial in designs of plasmonic devices that operates at ultraviolet-visible wavelengths.