Rigidity for piecewise smooth homeomorphisms on the circle

We find conditions for two piecewise C2+ν homeomorphisms f and g of the circle to be C1 conjugate. Besides the restrictions on the combinatorics of the maps (we assume that the maps have bounded combinatorics), and necessary conditions on the one-side derivatives of points where f and g are not differentiable, we also assume zero mean-nonlinearity for f and g.