Two characterisations of a random mean from a Dirichlet process, as a limit of finite sums of a simple symmetric form and as a solution of certain stochastic equations, are investigated. These are used to reach results on and new insight into such random means. In particular, identities involving functional transforms and recursive moment formulae are established. Furthermore, characterisations for several choices of the Dirichlet process parameter (leading to symmetric, unimodal, stable and finite mixture distributions) are provided. The theory also extends to the case of several random Dirichlet means simultaneously, a situation not covered earlier in the literature.