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Your question translates into the language of orbifolds as saying: what is known about spherical $n$-orbifolds with underlying space homeomorphic to $S^n$?

In $S^2$, the examples you give are all there are.

Orbifolds with the geometry of $S^3$ were enumerated by William Dunbar in his thesis. His published paper MR1118824 contains the enumeration of the 21 oriented spherical 3-orbifolds $S^3$-orbifolds which do not have a circle fibration over a 2-orbifolds2-orbifold. The equivalence relation here is up to orientation preserving isometry; if you allow orientation reversing isometry then the list is cut down somewhat. Each of the 21 has underlying space homeomorphic to $S^3$. At the end of Dunbar's paper you will see that exactly 8 of the 21 are orientable double covers of Coxeter groupsgroup quotients, with the corresponding Dynkin diagrams listed out explicitly. That leaves 13 examples as you ask for in $S^3$.

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Your question translates into the language of orbifolds as saying: what is known about spherical $n$-orbifolds with underlying space homeomorphic to $S^n$?

In $S^2$, the examples you give are all there are.

Orbifolds with the geometry of $S^3$ were enumerated by William Dunbar in his thesis. His published paper MR1118824 contains the enumeration of the 21 oriented spherical 3-orbifolds which do not have a circle fibration over a 2-orbifolds. The equivalence relation here is up to orientation preserving isometry; if you allow orientation reversing isometry then the list is cut down somewhat. Each of the 21 has underlying space homeomorphic to $S^3$. At the end of Dunbar's paper you will see that exactly 8 of the 21 are orientable double covers of Coxeter groups, with the corresponding Dynkin diagrams listed out explicitly. That leaves 13 examples as you ask for in $S^3$.