Interchanging zero and infinity in R^3 by the inversion map

(x,y,z) ⟶ (x,y,z)/(x^2+y^2+z^2)

turns a torus which is centered at zero inside out. This can be done continuously,
if we think of the torus in R^3 as the stereographic image of a torus in S^3. In R^3
we can rotate a plane around one of its lines by 180 degrees - so that we can see the
other side of the plane. Similarly, in S^3 we can rotate a Clifford torus around one
of the great circles on it also by 180 degrees and this exchanges the two sides of the
torus. If we follow this rotation by stereographic projection into R^3, then we can
actually see it. The rotating torus in S^3 has to pass through the center of the
stereographic projection - at this moment the projection into R^3 looks like a plane
with a handle.