Univariate continuous distributions: Symmetries and transformations

If the univariate random variable XX follows the distribution with distribution function FF, then so does Y=F−1(1−F(X))Y=F−1(1−F(X)). This known result defines the type of (generalised) symmetry of FF, which is here referred to as T-symmetry; for example, ordinary symmetry about θθ corresponds to Y=2θ−XY=2θ−X. Some distributions, with density fSfS, display a density-level symmetry of the form fS(x)=fS(s(x))fS(x)=fS(s(x)), for some decreasing transformation function s(x)s(x); I call this S-symmetry. The main aim of this article is to introduce the S-symmetric dual of any (necessarily T-symmetric) FF, and to explore the consequences thereof. Chief amongst these are the connections between the random variables following FF and fSfS, and relationships between measures of ordinary symmetry based on quantiles and on density values.