This paper considers structural non-parametric random utility models for continuous choice variables with unobserved heterogeneity. We provide sufficient conditions on random preferences to yield reduced-form systems of non-parametric stochastic demand functions that allow global invertibility between demands and non-separable unobserved heterogeneity. Invertibility is essential for global identification of structural consumer demand models, for the existence of well-specified probability models of choice and for the non-parametric analysis of revealed stochastic preference. We distinguish between new classes of models in which heterogeneity is separable and non-separable in the marginal rates of substitution, respectively.