  An early result of Noncommutative Geometry was Connes' observation in the
1980's that the Dirac-Dolbeault cycle for the $2$-torus $\mathbb{T}^2$, which
induces a Poincar\'e self-duality for $\mathbb{T}^2$, can be 'quantized' to
give a spectral triple and a K-homology class in $KK_0(A_\theta\otimes
A_\theta, \mathbb{C})$ providing the co-unit for a Poincar\'e self-duality for
the irrational rotation algebra $A_\theta$ for any $\theta\in
\mathbb{R}\setminus \mathbb{Q}$. This spectral triple has been extensively
studied since. 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-trivial element $g$ of the modular group, a finitely generated projective
module $\mathcal{L}_g$ over $A_\theta \otimes A_\theta$ by using a
reduction-to-a-transversal argument of Muhly, Renault, and Williams, applied to
a pair of Kronecker foliations along lines of slope $\theta$ and $g(\theta)$,
using the fact that these flows are transverse to each other. We then compute
Connes' dual of $[\mathcal{L}_g]$ for $g$ upper triangular, and prove that we
obtain an invertible in $KK_0(A_\theta, A_\theta)$, represented by what one
might regard as a noncommutative bundle of Dirac-Schr\"odinger operators. An
application of $\mathbb{Z}$-equivariant Bott Periodicity proves that twisting
the module by the family gives the requisite spectral cycle for the unit, thus
proving self-duality for $A_\theta$ with both unit and co-unit represented by
spectral cycles.
