In this work, we describe an electro-optic material capable of orthogonally switching the polarization of the localized surface plasmon resonance scattering of single gold nanorods, independent of their orientation. Liquid crystal samples are prepared in a sandwich configuration with electrodes arranged so that an applied voltage induces alignment-switching of the liquid crystal molecules covering individual gold nanorods. Due to the birefringence of the nematic liquid crystal, the reorientation in the nematic director alignment causes a change in the output polarization of the scattered light. We propose the underlying mechanism to be based on a homogeneous nematic to twisted nematic phase transition and provide support for it via Jones calculus by modelling the effect of ideally aligned homogeneous nematic and twisted nematic phases on polarized light transmitted through the sample. In the model, we include the effects of sample thickness and surface plasmon resonance wavelength, expressed in terms of the phase retardation, χ, on the observed output polarization. We find four distinctively different trends for the output polarization as a function of the incident polarization as χ is varied. Two of these cases provide reproducible orthogonal polarization switching of the surface plasmon resonance while maintaining a high degree of polarization. These results are verified experimentally with liquid crystal cells of different thicknesses. The deviation of the experimental samples from ideal behaviour can be explained by the inherent variations in the surface plasmon resonance maximum and local cell thickness.