How It Works
The needle turns to point at a nearby charged object due to charges induced in the ends of the needle by the external charge, through electrostatic induction. For example, if a positively charged object is brought near, the mobile negative charges in the metal will be attracted to it, and move to the end of the needle nearest the object. The attractive force on these negative charges will then turn the needle until the end is nearest to the charged object, when it will stop. Conversely the positive charges in the needle will be repelled, and move to the far end of the needle. The repulsive forces will then push this end of the needle as far away from the object as possible. The result, after the needle stops swinging, is that the axis of the needle points through the object.
Either end of the needle can be attracted to the charged object; whichever happens to be nearest will turn to point at it. So the two ends of the needle are symmetric as far as its action is concerned. The versorium needle also responds identically regardless of the polarity of the attracting charge, so it cannot distinguish between a positive and a negative charge, unlike a compass needle, which has a "North" and "South" end which can distinguish between the "North" and "South" pole of a magnet.
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