Normal scale mixtures and dual probability densities

Let p be a probability density function representable as a scale mixture of normal probability densities. Whenever p is bounded, there is a dual density (p) over cap proportional to the characteristic function of p. We present numerous examples for pairs of dual probability densities representable as normal scale mixtures, among them generalized hyperbolic, Student's t and McKay's Bessel function, symmetric stable and exponential power, generalized logistic and generalized hyperbolic secant, alpha-Laplace or Linnik and generalized Cauchy densities.