Do electrons attract to each other like permanent magnets?
Not really. The repulsion from their electric charge overwhelms any magnetic attraction in most circumstances. There may be some very specific circumstances where this comes into play, but for the most part the magnetic field from their own spin has little effect on them.
As Drakkith mentioned, the magnetic moment of the electron is small enough that it never leads to a net attraction. However, it can definitely modulate the strength of the repulsion so that pairs of electrons aligned in different ways will have different energies.
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