We find all factorized duality functions for a class of interacting particle systems. The functions we recover are self-duality functions for interacting particle systems such as zero-range processes, symmetric inclusion and exclusion processes, as well as duality and self-duality functions for their continuous counterparts. The approach is based on, firstly, a general relation between factorized duality functions and stationary product measures and, secondly, an intertwining relation provided by generating functions. For the interacting particle systems, these self-duality and duality functions turn out to be generalizations of those previously obtained in [9] and, more recently, in [8]. Thus, we discover that only these two families of dualities cover all possible cases. Moreover, the same method discloses all self-duality functions for interacting diffusion systems such as the Brownian energy process, where both the process and its dual are in continuous variables.