Consider the orbit space $X/K$ with $X$ a symmetric space and $K$ a group. Let $x$ represent an orbit $Kx$ in $X/K$. Now let's introduce a subgroup $H \subset K$, split up $K$ into cosets $aH$ (with $a \in K/H$), and look at the parts of the orbit corresponding to each coset, i.e., $aHx$. It seems that the coset representative $a$ either rotates $Hx$ to a distinct subset of $X$ or leaves it be. In other words, are there any weird situations where only some points of the coset orbits $aHx$ and $bHx$ are the same but others are different?
I know of one famous example, the compound of five tetrahedra, in which five tetrahedra (corresponding to the five coset orbits $aTx$) together make up a dodecahedron $Ix$ (an orbit in $S^2/I$). So I was wondering what happens in the general/generic cases for $S^2$ and $H \subset K \subset SO(3)$.