An early result of Noncommutative Geometry was Connesโ observation in the 1980โs that the Dirac-Dolbeault cycle for the 2 2 -torus T 2 \mathbb {T}^2 , which induces a Poincarรฉ self-duality for T 2 \mathbb {T}^2 , can be โquantizedโ to give a spectral triple and a K-homology class in K K 0 ( A ฮธ โ A ฮธ , C ) \mathrm {KK}_0(A_\theta \otimes A_\theta , \mathbb {C}) providing the co-unit for a Poincarรฉ self-duality for the irrational rotation algebra A ฮธ A_\theta for any ฮธ โ R โ Q \theta \in \mathbb {R}\setminus \mathbb {Q} . Connesโ 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 b , a finitely generated projective module L b \mathcal {L}_{b} over A ฮธ โ A ฮธ A_\theta \otimes A_\theta by using a reduction-to-a-transversal argument of Muhly, Renault, and Williams, applied to a pair of Kronecker foliations along lines of slope ฮธ \theta and ฮธ + b \theta + b , using the fact that these flows are transverse to each other. We then compute Connesโ dual of [ L b ] [\mathcal {L}_{b}] and prove that we obtain an invertible ฯ b โ K K 0 ( A ฮธ , A ฮธ ) \tau _{b}\in \mathrm {KK}_0(A_\theta , A_\theta ) , represented by an equivariant bundle of Dirac-Schrรถdinger operators. An application of equivariant Bott Periodicity gives a form of higher index theorem describing functoriality of such โ b b -twistsโ and this allows us to describe the unit of Connesโ duality in terms of a combination of two constructions in KK-theory. This results in an explicit spectral representative of the unit โ a kind of โquantized Thom classโ for the diagonal embedding of the noncommutative torus.