I define a NICE space to be a connected Riemannian manifold $M$ such that for any two distinct points $p,q\in M$, there exists an isometry $R_{p,q}$ exchanging these two points (that is such that $R_{p,q}(p)=q$ and $R_{p,q}(q)=p$).
Every connected 2-point homogeneous space (defined by Hsien-Chung Wang) is clearly NICE.
Moreover, a cartesian product of NICE spaces is itself NICE.
Do you know if NICE manifolds have been studied and classified?
