A related question for $d=2$ has already been raised (and positively answered) at Math StackExchange here: Is it possible to obtain a sphere from a quotient of a torus? It is also trivially true for $d=1$, since $T^1=S^1$. I am wondering, whether this generalizes for higher dimensions as well.
Specifically, consider a torus $T^d$ constructed as a cube $[-1,1]^d$ with identified faces. Then additionally identify points $\boldsymbol{r}$ and $-\boldsymbol{r}$ within this cube to create an orbifold $T^d/\mathbb{Z}_2$. Is the result of such a procedure homeomorphic to sphere $S^d$?
For $d=2$ it can be easily shown that indeed $T^2/\mathbb{Z}_2\cong S^2$. The described procedure is equivalent to the one suggested by user126154 at the linked StackExchange question. It's also easy to visualize since both $S^2$ and $T^2$ are easily handled in 3 spatial dimensions.
But plain visualization fails for $d>2$ and I don't really know how to prove/disprove this kind of statements.
