Trying to follow up on Aaron's hints.
So one way to look at $X$ is as the product space $D^2 \times S^2$ with an equivalence relation $E$ on the boundary $\partial D^2 \times S^2 = S^1 \times S^2$.
The Equivalence relation $E$ on the boundary is : $({ b_1 \in S^1 }, { b_2 \in S^2 }) \sim ({ b_1 \in S^1 }, { Rb_2 \in S^2 })$
SO $X = (D^2 \times S^1) /E$.
Now consider the following subspace $Y$ of $X$:
$Y = { [-1,1] \times S^2 }$ with $ E_1 = (-1;y \in S^2) \sim (-1;Ry \in S^2)$ and $E_2 = (1;y \in S^2) \sim (1;Ry \in S^2)$ as equivalence relations on $Y$
$I=[-1,1]$ is a line passing through the origin of $D^2$. I think I could prove $Y$ to be homeomorphic to $S^3$.
Please check: This is probably a very crude way to show the homeomorphism, but this is the only way I know how to do this. May be there are some algebraic methods to prove this result ?
Visualize $Y$ as a combination two solid cylinders of $I \times D^2$. So I have two solid cylinders, $C_1$ and $C_2$, when joined along the cylindrical surfaces form $Y$. So each point on the cylindrical surface of $C_1$ has an equivalent point on the cylindrical surface of $C_2$, since they belong to the same sphere $S^2$.
The disks that belong to the top and bottom surfaces of cylinder $C_1$ are identified with the disks of cylinder $C_2$ due to equivalence relations $E_1,E_2$ on the boundaries of subspace $I \times S^2$ that it inherits from $E$. The relations are:
$E_1 = (-1;y \in S^2) \sim (-1;Ry \in S^2)$ and $E_2 = (1;y \in S^2) \sim (1;Ry \in S^2)$. \
Hence I have two cylinders $C_1$ and $C_2$ where any point on the surface of $C_1$ has one and only one equivalent point on the surface of $C_2$. A solid cylinders is homeomorphic to a solid ball, hence the subspace $Y$ is equivalent to two solid spheres where the surface points of one solid sphere are identified with surface points of the other solid sphere. This space is equivalent to $S^3$.
May be there is some way to get $S^4$ by extending the subspace $(I \times S^2)/[E_1,E_2]$ to the entire $(D^2 \times S^2)/E$ ??