Antipodal in the 2-torus

Question

I'm strugling to understandind the antipodal in the 2-torus. In this thread

Embedding of Klein bottle in $\mathbb{R}^4$ and Klein bottle as quotient

I find " the antipodal mapping $A:S\to S$ can be expressed as $(x, y)\mapsto(x+1/2,-y)$". But I don't understand since if I put for instance $x=0.7$ then $x+\frac{1}{2}\notin [0,1]$ in the same way $-y \notin[0,1]$ for any $y \in [0,1]$, but the parametrization of torus is defined in $[0,1]\times [0,1]$ according the thread above, So i don't know how fix this information.

Other doubt is: Why the autor of thread increasing the angle $4\pi$ in the function $G$.

Thanks in advance!