  We look at interval exchange transformations defined as first return maps on
the set of diagonals of a flow of direction $\theta$ on a square-tiled surface:
using a combinatorial approach, we show that, when the surface has at least one
true singularity both the flow and the interval exchange are rigid if and only
if tan $\theta$ has bounded partial quotients. Moreover, if all vertices of the
squares are singularities of the flat metric, and tan $\theta$ has bounded
partial quotients, the square-tiled interval exchange transformation T is not
of rank one. Finally, for another class of surfaces, those defined by the
unfolding of billiards in Veech triangles, we build an uncountable set of rigid
directional flows and an uncountable set of rigid interval exchange
transformations.
