How to group two halves of a circle in order to create a curve on its surface

I haven't played much with the "Style" functionality, but it allowed me to do a "drop curve" on a sphere and get both halves by selecting "Quilt" (instead of surface) from my "Query Select" pop-up.

Depending on the desired results you could use COS by Intersect which would allow you to select multiple references for the First and Second surfaces or planes. If you are looking to create something like a spiral you could create a helical surface and try the COS by Intersect. If you plan on using the COS option you need to create curves on each surface, snap the end point of one curve to boundary of the surface it lies on using the SHIFT key, the square indicating the point will change to a circle indicating it is on the boundary curve, select the end point of the curve on the other surface and snap the end point to the end point of the other curve using the SHIFT key, the square will change to an X indicating the end points are joined. You can then move the point along the boundary, keeping the end points of the curves connected, as well as apply end point conditions, free, normal, tangent, curvature.