  For the flip action of $\mathbb{Z}_2$ on an $n$-dimensional noncommutative
torus $A_\theta,$ using an exact sequence by Natsume, we compute the K-theory
groups of $A_\theta \rtimes \mathbb{Z}_2$. The novelty of our method is that it
also provides an explicit basis of $\mathrm{K}_{0}(A_{\theta}\rtimes
\mathbb{Z}_2),$ for any $\theta.$ As an application, for a simple
$n$-dimensional torus $A_{\theta},$ using classification techniques, we
determine the isomorphism class of $A_{\theta}\rtimes \mathbb{Z}_2$ in terms of
the isomorphism class of $A_{\theta}.$
