In $SO(3)$, if the 3 Euler angles are given by $(\phi,\theta,-\phi)$, what is the resulting space named?

Question

The space $SO(3)$ can be parameterized by 3 Euler angles $(\phi,\theta,\psi)$. If the 3 Euler angles are given by $(\phi,\theta,-\phi)$, what is the resulting space named? The convention here is $R(\phi,\theta,\psi)=R_z(\phi)R_y(\theta)R_z(\psi)$, where $R_z(\phi)$ represents the rotation around $z$ axis etc.

Answer 1

It seems that the space generated by these Euler angles when $\phi\in[0,2\pi), \theta\in[0,\pi)$ is a sphere with boundaries, since for any vector $x\in S^2$ the image under the rotation above is still $S^2$ minus a circle belonging to a plane containing $x$. This space looks like an open Poke-ball (the image attached has an exaggerated cutoff $\theta\in[0,\pi-0.05])$ and it has the topology of the complement of an annulus on a plane.