As an addition to the diagram helpfully posted by Old Pueblo, here's a modern photo showing rein-guides, which are the rings on the thick black leather strap/harness in the photo. They are for wagon/coach/buggy-pulling horses' harness. The terrets/rein-guides keep the long reins from drooping down along the horse's sides, where the reins could snag on something. The "swinger" terrets, like you found, are a fancier version of the simple ring terrets/rein-guides.
The other photo shows simple rings and another variety in the middle, in which the ring has an opening, for ease of inserting the "bearing rein" seen in Old Pueblo's diagram.