In intuitionistic fuzzy sets, there is a natural symmetry between degrees of truth and falsity. As a result, for such sets, natural similarity measures are symmetric relative to an exchange of true and false values. It has been recently shown that among such measures, the most intuitively reasonable are the ones which are also symmetric relative to an arbitrary permutation of degrees of truth, falsity, and uncertainty. This intuitive reasonableness leads to a conjecture that such permutations are not simply mathematical constructions, that these permutations also have some intuitive sense. In this paper, we show that each such permutation can indeed be represented as a composition of intuitively reasonable operations on truth values.