The general solution for modes in an asymmetric planar waveguide with an isotropic chiral core is given in terms of a pair of parameters related to the eccentricity of the polarization ellipse for the transverse electric field. This formulation provides insight into the transition, with increasing chirality of the core, from TE/TM modes to right-handed (RHC) and left-handed (LHC) circular polarization modes. The mode properties as a function of waveguide thickness and of frequency are discussed in detail. In addition to the usual cutoff at a minimum thickness (and maximum wavelength), left-handed elliptical modes are found to experience a cutoff at a mode-dependent maximum thickness (and minimum wavelength). The limiting case of a symmetric waveguide is also discussed.