Taking $T^2 = \mathbb{R}^2/\mathbb{Z}^2$, we see there is an involution $(x,y) \mapsto (-x,-y)$ which has 4 fixed points; $(0,0), (1/2,0), (0,1/2), (1/2,1/2)$. Indeed, on $T^n$, the map $v \mapsto -v$ has $2^n$ fixed points.
Now, if I'm not mistaken, for $n=2$, quotienting by this $\mathbb{Z}_2$ action gives a branched covering map $T^2 \to S^2$ which has 4 branched points (the fixed points). Away from the branched points, the map is a deg 2 covering. However, for $n=1$, quotienting by this action gives a branched cover of a closed interval.
Are there any $n \neq 2$ such that there is a branched double cover $T^n \to S^n$? In particular, it would be quite interesting if there is such a map which uses this involution action somehow.