Consider the torus $T=\{(x, y, z)\in \mathbb{R}^3| (\sqrt{x^2+y^2}-2)^2+z^2=1\}\subset \mathbb{R}^3$ and the action of $\mathbb{Z}_4$ on $T$ given by $\hat{a}+(x, y, z)=\left(\cos\left(\frac{a\pi}{2}\right)x-\sin\left(\frac{a\pi}{2}\right)y, \sin\left(\frac{a\pi}{2}\right)x + \cos\left(\frac{a\pi}{2}\right)y, z\right)$. I am asked to describe the quotient manifold $T/\mathbb{Z}_4$ (well, I am asked to prove first that the action is free and properly discontinuous, but that is easy since we are acting by a finite group and I only need to check for fixed points, which is really simple in this case).
I believe that this is a Klein bottle. Let me explain my reasoning. What our action does is basically to encode a rotation matrix in the $xy$-plane. Trying to picture this in my head, what I get by gluing the points accordingly is nothing but a Klein bottle. I know this is no way a rigorous proof. I would also be interested in that, but first and foremost I want to know if my heuristic answer is correct.