Summary. Auditory localization experiments are conducted to evaluate human ability to locate the position of a source of sound, and to determine how population characteristics might affect this ability. These experiments generate data that are circular, bimodal and repeated, and have hypothesized symmetry patterns that should be included and tested within the modelling framework. We propose a two-part mixture of wrapped Cauchy densities for these bimodal angular data, with random effects to model correlation between repeated measures. The effects of signal position and types of symmetry in the signal response around the circle are modelled by using circular B-splines. The model is used to investigate the effects of age and hearing impairment on the ability to localize a low frequency signal.