:You won't damage anything if you hook them up with polarity reversed from that recommended by the headphone maker, but you may notice a difference in sound quality or volume.
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For AC-coupled outputs feeding headphones, either via transformer or capacitor, polarity doesn't matter at all.
1. How do you know the correct polarity?
2. Most of my old 'phones pre-date Alnico
Rich
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:Polarity only matters if DC running through the phones. Many early radios applied +90V to one side of the 'phones with the other side running to the audio output tube's plate, providing the DC bias needed to operate the tube. (Certainly not a great scenario if one was operating a radio with a sweaty head while sitting on a grounded metal chair!) Since the 'phones had permag fields, (notionally at least) an incorrect direction of DC current could act against the static magnetic field, affecting the sound quality and (perhaps) even tending to demagnetize the weak pre-ALNICO magnets used back then.
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:For AC-coupled outputs feeding headphones, either via transformer or capacitor, polarity doesn't matter at all.
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I'm pretty certain that headphone (or horn-speaker driver) polarization ceased to be of much interest after the Kellogg-Rice speaker technology took hold and wiped out all of the competition almost overnight in the very late 1920s.