The speed events at shows where I ride (tiny, local schooling shows, mostly) require that riders dismount and lead their horses out of the arena after their run. I kind of figured that that's why some barrel reins had the snap on one side, to lead easier after running the pattern.
I love my nylon barrel reins, though. Easy to handle and I don't have to worry about dropping one split rein on a trail ride. Mine have snaps on both ends, so they double as a lead for my trail bridle.
A stubborn horse walks behind you, an impatient one in front of you, but a noble companion walks beside you ~ Unknown