Whoa... you actually meant a folding helmet.

I wouldn't use something that's not a bike helmet. The types of impacts they're tested for may be very different. For example, I understand that one thing that bike helmets are tested for is sliding on road surfaces without snagging (which could break the wearer's neck). Climbing helmets may be more similar to skate/stunt helmets, because I imagine they'd be subject to more direct impacts (such as falls, or falling objects). Unlike skate helmets, they may not be tested for some of the kinds of horizontal falls and skids that wheeled travel entails. Similarly, this is why you're not supposed to wear a hockey helmet for biking.

Some kinds of sport helmets are interchangeable -- I used a bike helmet for riding a horse when I was a kid, and you'll see that a lot in therapeutic riding programs too. I wouldn't use a horseback riding helmet on a bike, though, for the reasons I stated above.