Maybe a support arm for a threader?
I agree, it's part of a greater assembly. The smooth rod is a pivot and installs into another part. The clamp jaw (lower) is there, but the upper jaw might have inserted into the hollow shank of the screw. It might be offset to allow the missing jaw to exert more force opposite the direction of torque.
This, just from looking at it. I did a quick search of Ridgid's website, but couldn't see anything on current models of reamers or cutters or threaders that had that part on it.
HOWEVER, if you REALLY want to know, go to their website and submit a query to them with your pictures. I'm sure they have somebody there that knows....