We investigate the properties of a process where the subsequent values assumed by the state of a chaotic map are summed to each other and the result is constrained within a finite domain by a folding operation. It is found that the limit distribution is always uniform, that the folded sums tend to be independent of the future evolution of the chaotic trajectory and that, whenever the map state is multi-dimensional, the folded sum vectors tend to be made of independent components. As an example, an application to the formal derivation of the spectrum of chaotically frequency modulated signals is also reported.