Write $S^n$ for the $n$-dimensional sphere, the space of vectors of length $1$ in $(n+1)$-dimensional Euclidean space. Consider the antipodal action on $S^n$, i.e. the action of $\mathbb{Z}_2$ given by $x \mapsto -x$ for any $x \in \mathbb{S}^n$. Then the orbit space $S^n/\mathbb{Z}_2$ is the $n$-dimensional projective space $\mathbb{R}P^n$.
What is the orbit space of the action on $S^n \times S^n$ given by $(x,y) \mapsto (-x, -y)$? An uneducated guess would be $\mathbb{R}P^n \times \mathbb{R}P^n$, but this clearly is not true, because the fundamental groups do not agree. Perhaps the answer is $S^n \times \mathbb{R}P^n$? But I don't really see why that would be the case.
This quotient is a sphere bundle over $RP^n$: You can think of this construction as a special case of an associated fiber bundle, associated with the representation $$\pi_1(RP^n)\to O(n+1)=Isom(S^n)$$ sending the generator to the matrix $-I$. This bundle is non-orientable if $n$ is even, hence, the bundle is nontrivial for even $n$. The bundle is orientable for $n$ odd and I have to think if it is trivial or not. My guess is that it is trivial.