Involutions of a torus $T^n$. Let $T^n$ be a complex torus of dimension $n$ and $x \in T^n$. We have a canonical involution $-id_{(T^n,x)}$ on the torus $T^n$. I want to know for which $y \in T^n$, we have $-id_{(T^n,x)}=-id_{(T^n,y)}$ as involutions of $T^n$. 
My guess is, such $y$ must be a 2-torsion point of $(T^n,x)$ and there are $2^{2n}$ choices of such $y$. Am I right? 

You are basically right, at least in the case of elliptic curves. I will try to tackle it using the theory from the chapter on curves from Hartshorne's "Algebraic Geometry". Thus, I will denote the torus $T$ by $E$, as is more appropriate. 
Recall that if we choose a point $x \in E$, then we can consider the degree $2$ morphisms $\pi: E \rightarrow \mathbb{P}^{1}$ given by the linear system $|2x|$. This corresponds to a Galois extension $k(E) \ / \ k(\mathbb{P}^{1})$ of degree 2, therefore there is exactly one non-trivial element $\tau \in Gal(k(E) \ / \ k(\mathbb{P}^{1}))$, which is exactly the involution with the respect to the group structure on $(E, x)$, ie. $\tau = -id _{(T,x)}$ in your notation.
But $2$-torsion points on $(E, x)$ are exactly the branch points of $\pi$ and it follows that if $y$ is $2$-torsion, then the linear system $|2y|$ defines the very same map into $\mathbb{P}^{1}$. Thus, the inverse on $(E, y)$ corresponds to $\tau$ above by the same argument. That is, $- id _{(T,x)} = - id _{(T,y)}$. 
It is also clear that "$y$ is $2$-torsion on $(E,x)$" is a necessary condition for $- id _{(T,x)} = - id _{(T,y)}$ to hold. 
