I'm accustomed to reversible studs on my light stands, etc and hadn't thought of that on this rig. It was quite a number of years ago that I originally mated these. I'm attaching a picture of that part of the shaft. If the stud is reversed, would it then need the adapter as I'm using it on the head side to attach to the tripod shaft top?
If the screw is indeed reversible (which i think it should be) you never need an adapter. If the thing you want to put on the column requires either 1/4 or 3/8 thread, you just put the screw in with the correct side up.