Why do electrons and positrons exhibit opposite helical motion in a magnetic field?

When you throw an electron through a solenoid, it moves helically around the field lines, as per this schoolphysics illustration:

Then if we were to throw a positron through the solenoid, it would also move helically, but "the other way". One could liken their paths to left-handed and right-handed screw threads. We can see these paths in bubble-chamber pictures like this one from the BC website which anna referred to in a previous answer:

Now, we can read about this sort of thing in various textbooks, such as this. And I'm sure we all know that the force acting on an electron is perpendicular to the magnetic field lines, and we can read about the Lorentz force and the right hand rule. But what we never seem to see is why the electron and positron move the way that they do. Saying "they move like they do because of the force on them" doesn't explain anything at all. It's a non-answer. Can anybody explain why there's this rotational force, and why it rotates the electron path one way and the positron path the other? Why do electrons and positrons exhibit opposite helical motion in a magnetic field?

But what we never seem to see is why the electron and positron move the way that they do. Saying "they move like they do because of the force on them" doesn't explain anything at all. It's a non-answer.

The equation of motion for charge particle (electron,positron) in magnetic field is

$$m\frac{d}{dt}\left(\frac{\mathbf v}{\sqrt{1-\frac{v^2}{c^2}}}\right) = q\mathbf v \times \mathbf B(\mathbf r,t)$$

where $\mathbf r$ is position of the particle, $\mathbf B(\mathbf r, t)$ is magnetic induction of external field at this position and time, $q,m$ are charge and mass and $\mathbf v$ is velocity of the particle.

For uniform $\mathbf B$, this equation has solutions that describe helical motion, in agreement with observations. This is an explanation of the helical trajectories; circular motion is a special case of this helical motion.

Can anybody explain why there's this rotational force, and why it rotates the electron path one way and the positron path the other?

Electron has electric charge $q=-1.6\times 10^{-19}$ C (by convention, electron is ascribed negative charge). Positron is ascribed $q=1.6\times 10^{-19}$ C. It is this difference in sign which leads to opposite directions of magnetic force. Imagine electron and positron far from each other, having the same velocity in the same uniform field. Since the magnetic forces acting on the two particles have the same magnitude but opposite directions, the particles will deflect with same rate but to opposite directions. Thus the helices they follow are left-handed and right-handed.

There are two factors at play here.

1. The Lorentz force which causes the paths to bend with a radius proportional to the particles velocity and with a sense that dependent on both the particles charge and the direction of the particles velocity. In high energy (compared to $m_e$ events) such as the one pictured, the particles are nearly co-linear at the start. Note that there is nothing special about leptons (electrons and positrons) this way, other charged particles also experience this force and obey the same set of rules.

2. The energy loss suffered by the particles in the detector medium (PDF-link, I'm afraid) which causes their momentum to fall steadily. This explains the spiral rather circular nature of the observed tracks.

The electron has three well known properties, its electric charge, its magnetic dipole moment and its intrinsic spin. All three are constant quantities. And to prevent contradition about the reality of this intrinsic spin, it was shown in the Einstein-de-Haas experiment, that this spin really has to do with a rotation of the electron.

It has to be stated, that the magnetic dipole moment and the intrinsic spin in the electron are aligned. This is a very important fact for the following explanation.

Being under the influence of a magnetic field, the electron's magnetic dipole moment get aligned. If the electron is not moving or if the electron moved parallel to the magnetic field that is it, nothing more happens.

But if the electron moves non parallel to the external magnetic field there came in the game the electron intrinsic spin. Due to the torque induced precession (gyroscopic effect) a rotating body tries to resistant its deflection. One can feel this by deflecting a rotating wheel from a bicycle. The magnetic field align the magnetic dipole moment and this time the intrinsic spin too. The spin resists against the deflection and emitting photons go back in the direction of his previous state. This repeats many times until the electron comes to rest. So the sketch in your question is not complete. The electron moves in tangerine slices and of course the path is a spiral and ends with the electron in rest.

The positron has the same values of the magnetic dipole moment and the intrinsic spin, but the direction of the spin in relation to the direction of the magnetic dipole moment is opposite to this relation of electrons. This is the reason, the "why", you are asking for.