If one can climb up then one can climb left, climb right, and climb down. The verb "climb" may mean to ascend, but that is not really relevant. The reason is because it is not how people use language. (And how people use language has far more of an impact than what a dictionary says.)
"Climb" and "climb up" seem to be two different verbs to me. One, I can climb a ladder and I can climb up a ladder.
I can climb a real ladder, the coroporate ladder, or the social ladder!
This is a similar phenomenon as "butter."
"I can purchase butter" - the noun
I can "butter my toast" - verb
and I can "butter you up" - verb
The last instance is a phrasal or particle verb. It is my contention that climb down is a particle verb that has been "invented" because we do not have a verb that means "climb down" that is in common usage. Sure, there is probably a Latinate verb out there that means "climb down" but they do not lack the power of our Germanic roots.
Always avoid Latinate verbs - they weaken your writing.