This paper investigates the problem of conservation of combinatorial structures in genome rearrangement scenarios. We characterize a class of signed permutations for which one can compute in polynomial time a reversal scenario that conserves all common intervals, and that is parsimonious among such scenarios. Figeac and Varré (WABI 2004) announced that the general problem is NP-hard. We show that there exists a class of permutations for which this computation can be done in linear time with a very simple algorithm, and, for a larger class of signed permutations, the computation can be achieved in subquadratic time. We apply these methods to permutations obtained from the X chromosomes of the human, mouse and rat.