The concept of rotation symmetric functions from the Boolean domain is extended to the multiple-valued (MV) domain. It is shown that symmetric functions are a subset of the rotation symmetric functions. Functions exhibiting these kinds of symmetry may be given a compact value vector representation. It is shown that the Reed-Muller-Fourier spectrum of a function preserves the kind of symmetry and therefore it may be given a compact vector representation of the same length as the compact value vector of the corresponding function. A method is presented for calculating the RMF spectrum of symmetric and rotation symmetric functions from their compact representations. Examples are given for 3-valued and 4-valued functions.