Extract issue..

[micheus]

How do you get, once you have the extraction, from a "flat" single surface, back to a solid object, with the same curvature on both sides?

Finally worked out a way. Had to extract "twice", along normals, once out to 0.2, a second time out to 0.19. I inverted the second one, then selected the "blank" sides (i.e., the parts missing geometry), and bridged them.

Oh God, what are you doing?
Use the steps I show you in the image and you will get this easily.

Oh, found a way around the "twisting" effect of circularize too - just make a sphere that is the right number of sides to begin with (duh!).

But, for the sake of understanding what I meant with twisting (adding extra cuts, then edge loop + circularize on each of the new circumferences), I will post an image.

You really seems to like the hard way.
Select the edges loops you want to twist, then change to vertex mode [V] and in the context menu look for Deform->Twist!