@marc.a.grossman, looks like your screencast didn't make it. However, I think I get the picture from your description. It's worth a little explanation about what Redefine Sketch Plane does and doesn't do.
First, let's address your last question first: what is green geometry? Green indicates that the geometry is "fixed". It's the same as if you just draw a line, then apply a Fixed constraint to it. Purple indicates a Projected geometry - for instance if you Project an edge from the model into your sketch. Projected geometry is totally driven by the object that was projected. So, if the edge moves, the projected purple line will move, as well. When you Redefine Sketch Plane for a sketch, all Projected geometry is converted into Fixed geometry. The reason for this is that Redefine has to account for all possible planes that are selected for new geometry. There is no guarantee that the edges projected into your current sketch will have any meaning, or logical placement, in the new plane. Constraints to that geometry would be unpredictable. If Fusion kept these as Projected geometry, the chances are good that your sketch would completely blow up if placed on another plane. So, to prevent that, Fusion converts all Projected geometry into Fixed geometry, leaving your sketch more or less intact.
As to the flipping about one axis, that is a matter of sketch coordinate system. If you pick a plane where the X axis for the new sketch is rotated or flipped as compared to the X axis of the original sketch, then yes, the sketch geometry will flip or rotate to be in line with the new sketch coordinate system.
For myself, if I find I need to do this, it's almost always easier to create a new sketch, copy/paste the non-projected geometry from the original sketch into the new one, then edit features to consume the new sketch. You can make Redefine work, but sometimes it is more painful than it's worth.
Jeff Strater
Engineering Director