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Spectra for non-selfadjoint operators often display fascinating features, from lattices of eigenvalues for operators of Kramers-Fokker-Planck type to eigenvalues for operators with analytic coefficients in dimension one, concentrated to unions of curves. In this talk, we shall discuss spectra for non-selfadjoint perturbations of selfadjoint semiclassical operators in dimension 2, assuming that the classical flow of the unperturbed part is completely integrable. We give complete asymptotic expansions for all individual eigenvalues in suitable regions of the complex spectral plane, close to the edges of the spectral band. It turns out that those eigenvalues have the form of the "legs in a spectral centipede" and are generated by suitable rational flow-invariant Lagrangian tori. This is joint work with Johannes Sjostrand.