I've always heard of it referred to as "Devil's Highway" rather than Devil's Rope. Due to it being highway # 666 before getting renumbered due to lobbying from religious folk.
But I love that ride. I've only done it from north to south, from Hannigan's Meadow and Alpine down to Clifton/Morenci.
Great elk/antelope/deer hunting area, as long as you have the stamina to hike it. You ain't gonna be able to just jump out of your truck and shoot a critter and haul it 50 feet to the bed of the truck back there.
Those golden plains just under the rim climb are hypnotizing.
ETA: A google search of "Devil's Rope" suggests that the term applies to barbed wire. This highway, however, is hit #1 when googling for "Devil's Highway Arizona" and in the top 3 when googling "Devil's Highway."