When one is climbing a road, they are ascending. When it comes to roads, the words climbing and ascending are synonymous. When one climbs a mountain, one is also ascending, but one is not on a road. Use of hands and feet, and climbing tools may be required, and one can climb down. Climbing up, and ascending are synonymous in that instance, as are climbing down, and descending.
But if you are on a road, climbing down and descending are no longer synonymous. You would hear descending. You would not hear "climbing down". I don't know technically why, but I do know you would likely not hear "climbing down" in conjunction with forms of travel on a road. You might see usage that the 'road' climbs down, but even that would typically be stilted and unusual.
While one can climb up a road, one cannot climb down a road. You would be descending, rather than climbing down.
Edit: After consideration, I think I understand WHY one would not "climb down" a road. I went to dictionary.com, and got these:
verb (used without object)
1. to go up or ascend, especially by using the hands and feet or feet only:
to climb up a ladder.
to proceed or move by using the hands and feet, especially on an elevated place;
Climbing can be ascending, which is why we can climb a road up the hill. Or it can be moving using the hands and feet, which would be the usage example "I climbed up the mountain." Again, using the hands and feet, one could climb back down the mountain - or the tree, or whatever one did the Spiderman routine to ascend. But when one removes the hand and foot part of the action, one is limited to ascending as the meaning. And one does not ascend down. Thus, one would not climb down the road.
However, I can not refrain from noting that an artistic and clever writer might be able to use climb down in just this way to somehow provide a more evocative sentence. But we won't find it in everyday usage.