Both existing models for non-symmetric distributions on 3-dimensional rotations and their associated one-sample inference methods have serious limitations in terms of both interpretability and ease of use. Based on the intuitively appealing Uniform Axis- Random Spin (UARS) construction of Bingham, Nordman, and Vardeman (2009) for symmetric families of distributions, we propose new highly interpretable and tractable classes of non-symmetric distributions that are derived from mixing UARS distributions. These have an appealing Preferred Axis-Random Spin (PARS) construction and (unlike existing models) directly interpretable parameters. Non-informative one-sample Bayes inference in these models is a direct generalization of UARS methods introduced in Bingham, Vardeman, and Nordman (2009), where credible levels were found to be essentially equivalent to frequentist coverage probabilities. We apply the new models and inference methods to a problem in biomechanics, where comparison of model parameters provides meaningful comparisons for the nature of movement about the calcaneocuboid joint of three different primate subjects.