Cohomology of $\mathcal A_\theta^{alg} \rtimes \mathbb Z_2$ and its Chern-Connes pairing

We calculate the Hochschild and cyclic cohomology of the $\mathbb Z_2$ toroidal orbifold $\mathcal A_\theta^{alg} \rtimes \mathbb Z_2$. We also calculate the Chern-Connes pairing of the even cyclic cohomology group with the known elements of $K_0(\mathcal A_\theta^{alg} \rtimes \mathbb Z_2)$.