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The limited slip is designed to lock the wheels together when one spins much faster than the other. Otherwise it is an open diff, such as during normal driving conditions or even slight turns. If you were to make a sharp turn in the snow and lose traction, once the tires start to spin they would lock together and hopefully help you regain traction.
This is why you said your tire on the pavement stayed stationary while the other one spun, and then broke loose and started to spin too.
Limited slip does not "lock" the wheels together. It uses a clutch system that engages after one tire slips too much--that's what allows a small amount of slippage when you go around turns on dry pavement. In that scenario, one tire travels quite a bit farther than the other, so there has to be some give to avoid spinning a tire or hopping. And, yes, lockers in snow are a bad thing, because one tire would be forced to lose traction in a turn, which can cause both to lose traction.
Here's a good explanation of the different differential systems:
I have herd that Limited slips rear ends require a special type of gear oil. Is this the case for Fords Limited slip rear ends?
Yes, you either need to add a friction modifier, or buy a gear oil that already has it. You'll know if you don't have enough becuase your rear end will make all sorts of noise when you go around turns and such.