In this article we study local rigidity properties of generalised interval exchange maps using renormalisation methods. We study the dynamics of the renormalisation operator $$\mathcal {R}$$ acting on the space of $$\mathcal {C}^{3}$$ -generalised interval exchange transformations at fixed points (which are standard periodic type IETs). We show that $$\mathcal {R}$$ is hyperbolic and that the number of unstable direction is exactly that predicted by the ergodic theory of IETs and the work of Forni and Marmi–Moussa–Yoccoz. As a consequence we prove that the local $$\mathcal {C}^1$$ -conjugacy class of a periodic interval exchange transformation, with d intervals, whose associated surface has genus g and whose Lyapounoff exponents are all non zero is a codimension $$g-1 +d-1$$ $$\mathcal {C}^1$$ -submanifold of the space of $$\mathcal {C}^{3}$$ -generalised interval exchange transformations. This solves a conjecture analogous to that of Marmi–Moussa–Yoccoz, stated for almost all IETs, in the special case of self-similar IETs.