Looks like a neck rope to me guys. If you were doctorig calves, and were alone, you would run your rope thru the neck rope and tie-off. This keeps your horse facing in the direction of you and what you roped. It also prevents the horse from turning and dragging what you roped away from you While you tie or doctor. In other words, it keeps the horse faced-up. But I could be wrong.
Martin that is what I thought it was also!!But I do not ride with either so did'nt want to sound stupid!!lol