This question falls under several categories {software, computer graphics, graphic design, art, cartography, geographic information, mathematical transforms}, but the software (specifically for Windows 10 [64-bit]) to perform this function is what I'm seeking.

I'm trying to shift geometric figures (specifically the borders of a political or geographic area) across a sphere:
Click image for larger version.  Name: Sleeve-transform on genus 0 topology.png  Views: 60  Size: 28.1 KB  ID: 108254
As you can see from the picture above, all points are meant to follow meridional paths, stretching the figure outward _around_ the surface's radial slices like an embedded rubber band until it crosses a great circle, then contracting back down to its original size a the far end of the sphere (so that the figure is now turned inside out).

Basically, it's a question of flipping a figure's borderline radial distances around its mean radius (everting dents and inverting bulges). I know what I want, but I don't know how to get there.

It's clearly not a rotation, translation (in the usual sense of being identical to a rotation), reflection (though that's close), scale, shear, or skew. The closest terminology that I've imagined so far is a "sleeve-transform on genus 0 topology", but I haven't found the right terms for a decent Google search on the underlying mathematical techniques with which to approach the transform.

Right now, I'm doing it by hand -- finding the inscribing and circumscribing circles, taking the centroid as a locus from which to measure the edge-coordinates, and flipping the annulus of circles around the mean annular radius. A pain in the ass, but functional for figures that contain their centroid within their boundary (simple polygons are easy, but not something complex like a country's border, and definitely no good for figures with an external centroid (e.g.: the majuscule letter "F", or a sans serif minuscule "f").