Two concepts of symmetry for the distributions of positive random variables Y are log-symmetry (symmetry of the distribution of logY) and R-symmetry . In this paper, we characterise the distributions that have both properties, which we call doubly symmetric. It turns out that doubly symmetric distributions constitute a subset of those distributions that are moment-equivalent to the lognormal distribution. They include the lognormal, some members of the Berg/Askey class of distributions, and a number of others for which we give an explicit construction (based on work of A.J. Pakes) and note some properties; Stieltjes classes, however, are not doubly symmetric.
"Distributions that are both log-symmetric and R-symmetric." Electron. J. Statist. 2 1300 - 1308, 2008. https://doi.org/10.1214/08-EJS301