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? 

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 $\mathbb C^n\to X=\mathbb C^n/\Lambda$. 
