Yes the OP is describing "The Devils Highway" which is widely regarded as the greatest motorcycle road in North America. I've ridden it on my bike half a dozen times and though I never counted it is reported to have 1,100 curves in just 95 miles and at least a quarter of those curves are first gear events... on a motorcycle. :S
The views are spectacular and most of the road is at 7 to 9,000'. One of the biggest challenges aside from all of the tight {and I mean very tight curves} is the relatively short weather window to safely ride the DH. Obviously winter is off the table but during summer the Monsoon thunderstorms {and this year is setting records almost daily} cause lots of rockslides rendering the road dangerous to impassable. Given the light traffic it gets keeping it clear is a pretty low priority for ADOT.
My bride and I first rode the DH in late May which was fine and early fall can be doable as well but whenever you go watch the weather and don't try it if there has been recent storm activity.
It would be fine in any car our truck but towing, especially a large rig, is not for the feint of heart or sound of mind. :B
Here isa shot taken about halfway between Morenci and Alpine looking west: