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We introduce the notion of self-similar actions of grouopids on other groupoids and Fell bundles. This leads to a new imprimitivity theorem arising from such dynamics, generalizing many earlier imprimitivity theorems involving group and groupoid actions. CONTENTS 1. Introduction 2. Self-similar actions 2.1. Self-similar left actions on groupoids 2.2. The self-similar product groupoid: A generalized Zappa-Szép product 2.3. Haar systems for self-similar left actions 2.4. Rehash (from left to right) 3. The Orbit Space 3.1. Self-similar Para-Equivalences 3.2. Haar Systems on quotients 4. Self-similar actions on Fell bundles 4.1. Self-similar left actions on Fell bundles 4.2. The self-similar product Fell bundle 5. The orbit Fell bundle from self-similar actions 6. The symmetric imprimitivity theorem for self-similar actions 7. Examples Appendix A. Exercises in Topology Appendix B. Cheat sheet B.1. The two-way actions of  and  B.2. The two-way actions of  and  B.3. Actions that are in tune B.4. The induced actions on quotients B.5. Self-similar actions on Fell bundles References

An early result of Noncommutative Geometry was Connes' observation in the 1980's that the Dirac-Dolbeault cycle for the 2-torus T 2 , which induces a Poincaré self-duality for T 2 , can be 'quantized' to give a spectral triple and a K-homology class in KK 0 (A θ ⊗ A θ , C) providing the co-unit for a Poincaré self-duality for the irrational rotation algebra A θ for any θ ∈ R ∖ Q. Connes' proof, however, relied on a K-theory computation and does not supply a representative cycle for the unit of this duality. Since such representatives are vital in applications of duality, we supply such a cycle in unbounded form in this article. Our approach is to construct, for any non-zero integer b, a finitely generated projective module L b over A θ ⊗ A θ by using a reduction-toa-transversal argument of Muhly, Renault, and Williams, applied to a pair of Kronecker foliations along lines of slope θ and θ + b, using the fact that these flows are transverse to each other. We then compute Connes' dual of [L b ] and prove that we obtain an invertible τ b ∈ KK 0 (A θ , A θ), represented by an equivariant bundle of Dirac-Schrödinger operators. An application of equivariant Bott Periodicity gives a form of higher index theorem describing functoriality of such 'b-twists' and this allows us to describe the unit of Connes' duality in terms of a combination of two constructions in KK-theory. This results in an explicit spectral representative of the unit-a kind of 'quantized Thom class' for the diagonal embedding of the noncommutative torus.