One can imagine $T^3$ much like he can imagine $T^2$: as a flat box with opposite faces identified. One may put coordinates on $T^3$, each of which would logically range from $0$ to $2\pi$.
To get $S^2$ from $T^2$, we think about throwing out half of $T^2$ and identifying two lines of points on $T^2$ as two poles on $S^2$. One can conceive of this process on the fundamental polygon of $T^2$ by chopping it in half and collapsing the two chopped edges as single points. Before identifying points, it is clear that after chopping, we have $S^1 \times I$, a cylinder of height $\pi$.
How about getting $S^3$ from $T^3$? According to the wikipedia article on the glome, all hyperspherical coordinates range from $0$ to $\pi$ except one, which ranges from $0$ to $2\pi$.
This leads me to believe that we should chop two rectangular prisms off of $T^3$.
Say we begin by chopping $T^3$ in half. Then an angle ranges from $0$ to $\pi$. What we have then is $S^1 \times S^1 \times I$, which I'm guessing may be some weird hypercylinder of height $\pi$. Now if we identify the two faces at $0$ and $\pi$ as one line, we get $S^1 \times S^2$, is this correct? Finally, to get $S^3$, we take a slice into $S^1 \times S^2$ to get $I \times S^2$, then we collapse opposite faces into lines to obtain $S^3$.
What about getting $\mathbb{R}P^3$ out of all this? Is there an easy way to see it?