Many parametric models can be enriched by introducing additional parameters through transmutation, mixing, or compounding techniques. In this paper, we develop the framework of doubly generalized transmutation models (DGTMs), obtained by the repeated application of rank transmutation maps and their generalizations. We show that several flexible families already available in the literature can be reinterpreted as instances of double or multiple transmutation, thus unifying apparently disparate constructions under a common perspective. A key feature of DGTMs is their ability to flexibly control symmetry through parameterization, enabling more accurate modeling of asymmetric or heavy-tailed phenomena. We also discuss the potential extension of these models to the bivariate case. In addition, we introduce the gentransmuted R package, Version 1.0, which provides routines for data generation, parameter estimation, and model comparison for generalized transmutation models. Two real data applications illustrate the practical advantages of this approach, highlighting improved model fit relative to classical alternatives. Our results underscore the value of transmutation-based methods as a systematic tool for generating flexible probability distributions and advancing their computational implementation.
Arnold et al. (2025) studied this question.