Summary: Based on a constructive representation, which distinguishes between a skewing mechanism \(P\) and an underlying symmetric distribution \(F\), we introduce two flexible classes of distributions. They are generated by nonparametric modelling of either \(P\) or \(F\). We examine properties of these distributions and consider how they can help us to identify which aspects of the data are badly captured by simple symmetric distributions. Within a Bayesian framework, we investigate useful prior settings and conduct inference through MCMC methods. On the basis of simulated and real data examples, we make recommendations for the use of our models in practice. Our models perform well in the context of density estimation using the multimodal galaxy data and for regression modelling with data on the body mass index of athletes.