A standard interval exchange map is a one-to-one map of the interval that is locally a translation except at finitely many singularities. We define for such maps, in terms of the Rauzy-Veech continuous fraction algorithm, a diophantine arithmetical condition called restricted Roth type, which is almost surely satisfied in parameter space. Let T0 be a standard interval exchange map of restricted Roth type, and let r be an integer ≥2. We prove that, amongst Cr+3 deformations of T0 that are Cr+3 tangent to T0 at the singularities, those that are conjugated to T0 by a Cr-diffeomorphism close to the identity form a C1-submanifold of codimension (g−1)(2r+1)+s. Here, g is the genus and s is the number of marked points of the translation surface obtained by suspension of T0. Both g and s can be computed from the combinatorics of T0.