If you ask around about magnetic fields, you will read seemingly-authoritative articles which say magnetism is a consequence of length contraction. This is widely taught and is repeated in answers such as this one which talks about the magnetic force between current-carrying wires. I'm sure we're all happy with this force, which you can see described in the picture from Rod Nave's hyperphysics:
However note the concentric magnetic field lines around the wire. We know that a charged particle such as an electron will circle around these field lines, as per this depiction courtesy of Chegg homework help:
In addition we know that a positron will circle the magnetic field lines the other way. If the electron path is a left handed helix, the positron path is a right-hand helix. Or a circle, as per these pictures of an electron beam in a uniform magnetic field.
The length-contraction explanation for two wires moving together sounds fairly plausible. However there doesn't seem to be any way that length contraction in that linear wire can result in the opposite circular motion of electrons and positrons. The so-called explanations I've found are woeful, little more than smoke and mirrors and a rabbit from a hat. Would anybody care to have a crack at explaining how this happens? There will be a 100-point bounty to the least-worst answer.
How can length contraction result in electron circular motion in a magnetic field, and the opposite circular motion for a positron?
Edit: for clarification, see this drawing:
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