Coherent scattering of an elliptically polarized tweezer into a cavity mode provides a promising platform for cooling levitated nanoparticles into their combined rotational and translational quantum regime [J. Schäfer, H. Rudolph, K. Hornberger, and B. A. Stickler, Phys. Rev. Lett. 126, 163603 (2021)]. This article presents the theory of how aspherical nanoparticles are affected by elliptically polarized laser beams, how two orthogonal cavity modes enable rotational and translational cooling, and how the resulting power spectra contain signatures of rotational nonlinearities. We provide analytic expressions for the resulting trapping frequencies, optomechanical coupling strengths, cooling rates, and steady-state occupations, and we study their dependence on the tweezer ellipticity.