Current algorithmic studies of genome rearrangement ignore the length of reversals (or inversions); rather, they only count their number. We introduce a new cost model in which the lengths of the reversed sequences play a role, allowing more flexibility in accounting for mutation phenomena. Our focus is on sorting unsigned (unoriented) permutations by reversals; since this problem remains difficult (NP-hard) in our new model, the best we can hope for are approximation results. We propose an efficient, novel algorithm that takes (a monotonic function f of) length into account as an optimization criterion and study its properties. Our results include an upper bound of O(fn lg2n) for any additive cost measure f on the cost of sorting any n-element permutation, and a guaranteed approximation ratio of O(lg2n) times optimal for sorting a given permutation. Our work poses some interesting questions to both biologists and computer scientists and suggests some new bioinformatic insights that are currently being studied.