Just a question about an interaction, nothing huge as the number of character builds that can effectively use both is limited, but it's a curiosity.
For the sake of it though I will be using an example character, I don't care if the build is bad or broken, this is purely an interaction question.
12th Level Drow Vigilante: Wildsoul (Arachnid)
Relevant Feats: Drow Nobility, Improved Drow Nobility, Greater Drow Nobility.
This character has Levitate at-will as a spell-like ability, and a climb speed of 30 ft. Can these two things interact?
For example, the standard action to cast the levitate and then move action to "jump" the 20 leviated feet and cling to the cieling? While levitating along a wall, would it half their climb speed or would the fact they have climb speed allow them to laterally move as normal, just unable to fall? If they had 3 limbs on the wall, would they be able to make attacks as normal without the penalty for levitating?
Obviously again, there are probably better ways to have both these things than the example character, and if you were being levitated by an enemy not your own spell I would likely assume it wouldn't matter. They could peel you off the ceiling with the levitate move action. Additionally the example character has levitate at-will, they could end the effect and start it up again as needed to negate any downsides. These aren't the question, just the interaction between the two. There's no specific literature on their interaction for RAW, it feels largely like something up to DM's discretion, but figured I'd ask in case I'm missing something?
I don’t think they officially do. At most of you were to “fall” climbing somehow you just wouldn’t fall.