Putting a bit more thought into this and reading through the older replies:
I still agree that turning should in no way be part of the interaction; it just doesn't make sense why this would be a special case where deassigners climb.
Climbers and Sliders have a mechanically similar counterpart which isn't a permanent skill: the Shimmier. It wouldn't get cancelled by this object since it's not permanent, which is technically consistent, but it does make me wonder if perhaps it *should* cancel because the skills are so closely related in function. But then why stop at cancelling Shimmiers -- why not cancel *any* skill, *including* the removal of permanents? Or, on the other hand, there's Simon's earlier suggestion of having the skill remover only remove the skill but not cancel it: so a climber would *finish* climbing the wall, but it wouldn't be able to climb again unless another climber was assigned. But -- this would cause a unique situation in which the climbing/sliding/floating/gliding/whatever state is decoupled from the status of actually having the skill. On the other hand, it's at least a simple rule.
The problem with potential turning is that it's a complicated rule: Usually, the deassigner won't turn lemmings, but in certain cases (which you need to memorize), the desassigner will cause a turn. Non-turning is a simple rule, but there's one "oh, I didn't think it would work that way but I guess that makes sense" exception you have to memorize with the shimmier ("it should cancel a shimmier too, right?" seems like a reasonable misconception even if logically, it's not a permanent so it doesn't get cancelled). If you cancel shimmiers too, then that's a different actual exception that you have to memorize. Non-cancelling (the permanent skill will finish its activity but can't be used again) or all-cancelling (including non-permanents) seems like the simplest rule.
I can't remember if it was an option in the experimental or not, and I could see the UI for it being a problem, but assuming you could find a reasonable way (maybe if you hover your mouse over it?), there's also the option of allowing arbitrary combinations of cancelled skills, which I actually think would be more interesting for puzzles than a simple deeassigner that deassigns everything (whether it affects non-permanents or not): you could imagine a puzzle where you're given multi-athletes and you have to navigate them through a maze of deassigners that requires retaining at least a few of the skills to navigate, so you have to figure out which ones you can afford to pass through.
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