I'll report on recent progress in rigidity theory for nonlinear interval exchange transformations corresponding to cyclic permutations. Such maps can be viewed as circle homeomorphisms with multiple break points. I shall discuss both recent results on renormalizations of such maps in case of one break point (joint with S. Kocic and A. Teplinsky), and extension to the multiple-break setting (based on work in progress with A. Teplinsky).