Recently, Consonni and Veronese have shown that the form of the standard conjugate distribution for the mean parameter μ of a univariate natural exponential familyFcoincides with that of the distribution induced on μ by the standard conjugate distribution for the canonical parameter if and only ifFhas a quadratic variance function. In this article we present significant extensions of this result, identifying conditions under which transformations of the canonical or mean parameter preserve the form of the standard conjugate family. Generalizations to the multivariate case are also considered, and results relating Jeffreys's prior to the standard conjugate family are presented. The variance function is seen to play an important role throughout.