From the Wikipedia article on projective planes:
[...] consider the unit sphere centered at the origin in $\mathbb{R}^3$. Each of the $\mathbb{R}^3$ lines in this construction intersects the sphere at two antipodal points. Since the $\mathbb{R}^3$ line represents a point of $\mathbb{RP}^2$, we will obtain the same model of $\mathbb{RP}^2$ by identifying the antipodal points of the sphere. The lines of $\mathbb{RP}^2$ will be the great circles of the sphere after this identification of antipodal points.
My question is:
When the construction of the real projective plane is essentially identifying antipodal points of the sphere, what is its analogue when identifying antipodal points of the torus?
It is not absolutely clear what antipodal points on a torus are. However, if you take a standard torus in $\mathbb{R}^3$, obtained by rotating a circle $(x-a)^2+z^2 = b^2$ with $a>b>0$ about the $z$-axis, and if you call $(-x,-y,-z)$ the antipodal point of $(x,y,z)$, I believe you get the Klein bottle.