Real projective space and $n$-sphere

Question

I know that $$\mathbb{R}^n\mathbb{P}\cong \mathbb{S}^n/{\pm 1}$$ Is there another equivalence relation apart from treating antipodal points as the same which we can quotient out from $\mathbb{S}^n$ to get another interesting space?

Answer 1

You can get lens spaces by quotienting out by actions of cyclic groups on odd-dimensional spheres (I never know if the term "lens space" is reserved for $S^3$ or if other odd-dimensional spheres count too)

Note that the antipodal action is the only nontrivial free action on even-dimensional spheres

Answer 2

One that I can think of is to quotient along a circle of the sphere, which should yield $S^n/S^{1} \simeq S^{n-1} \wedge S^{n-1}$