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? 
 A: Yes, you are right: here is a proof (I have taken the liberty of slightly modifying your notations). 
Let $X=\mathbb C^n/\Lambda$ be the complex torus obtained by dividing out $\mathbb C^n$ by the lattice $\Lambda\subset \mathbb C^n$   ($\Lambda \cong \mathbb Z^{2n}$). This torus is an abelian Lie group, and this gives it much more structure than a plain complex manifold.     
Such a torus admits of the involution $-id=\iota _0: X\to X:x\mapsto -x$, a holomorphic automorphism of the complex manifold $X$.
But for every $a\in X$ it also admits of the involution $\iota _a: X\to X:x\mapsto 2a-x$, which fixes  $a$.
Your question amounts to asking for which $a\in X$ we have $\iota_ a=\iota_0=-id$.
This means $2a-x=-x$ for all $x\in X$ or equivalently $2a=0$.
So, exactly as you conjectured,  the required points $a\in X$ are the $2^{2n}$ two-torsion points of $X$, namely the images of $\Lambda/2$, the half-lattice points,  under the projection morphism $\mathbb C^n\to X=\mathbb C^n/\Lambda$. 
A: 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. 
