Technically speaking, we are talking about the eversion of a sphere. I found this link. Interestingly, the article says that there is no way of turning a circle inside out.
A spherical inversion is different from eversion. In 3 dimensions, an inversion can be thought of as the reflection in a spherical mirror. It maps points on the sphere to themselves. Points inside the sphere are mapped to the outside of the sphere and vice versa.
I looked on the Web for a simple explanation but could not find one, so here is how it works. For any point p, let d be the distance from p to c, the center of the sphere. Consider the value d/r, where r is the radius of the sphere. For p outside the sphere, d/r > 1 and for p inside the sphere, d/r < 1. The effect of the spherical inversion is to move p along the ray joining p to c in such a way that the new distance d’ satisfies d’/r is one over d/r. If d/r was 3 then d’/r is ⅓.
An inverted sphere is the inversion of one sphere in another. In two dimensions, the circular inversion of a circle is a circle, provided that the circle does not pass through the circle of inversion. I would guess that in a similar fashion, the inversion of a sphere would be another sphere.