For $A,B \subset \mathbb{R}^n$, consider the equivalence relation $(a,b) \sim(-a,-b)$ on $A \times B$. I'm trying to see what the quotient spaces of $S^2 \times S^1$ and $S^1 \times S^1$ by $\sim$ look like.
For the torus, I considered the fundamental polygon representation and divided it in four squares. Since $\sim$ identifies each square with the reflected square by the midpoint, by cut and pasting, I ended up with just a torus. This makes sense since I know that $S^1 \backslash \sim = S^1$, but I'm not sure that it is correct. For $S^2 \times S^1$ I don't know where to begin, and would appreciate any hint.