Rigidity of quasisymmetric mappings between metric spaces is sometimes determined by the structure of the spaces. For example, a quasisymmetry from the Sierpinski gasket to itself is actuallly an isometry of the Euclidean space. We study the class of quasisymmetric mappings between horizontal self-affine carpets and show that this class is rather small. Our method is to go to the tangent level, use the structure of weak tangents to get information on the rigidity of tangent mappings, and then pull back some of this information to the original mapping. (Joint work with Antti Käenmäki and Tuomo Ojala.)