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Positive and negative spins are merely to indicate that the spin of the electrons are different and does not necessarily reflect their motion up and down or side to side. It just shows that they are opposite and therefore interfere with the magnetic fields around them differently.
Just to add on to previous replies, the spin magnetic quantum number was deemed necessary after the Stern and Gerlach Experiment, where the electrons were found to have two different spins (hence +1/2 or -1/2). It's important to note that no two electrons in the same atom have the same four quantum numbers.
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