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?