We describe a new and robust method to prove rigidity results in complex dynamics. The new ingredient is the geometry of the critical puzzle pieces: under control of geometry and \complex bounds", two generalized polynomial-like maps which admits a topological conjugacy, quasiconformal outside the lled-in Julia set are, indeed, quasiconformally conjugated. The proof uses a new abstract removability-type result for quasiconformal maps, following ideas of Heinonen