This description of the Stern-Gerlach experiment has the best pictures but omits some detail that had stumped me. I could see that a charged particle, spinning or not, is not pushed by a uniform magnetic field in the direction of the field. Chapter 35 in vol. 2 of the “Feynman Lectures on Physics” describes this experiment with more detail. The depicted shape of the magnet pole pieces is critical for it produces a non-uniform magnetic field thru which the electrons fly. Here is how a non-uniform field pushes a spinning charge:

A spinning charge is like a circulating current which produces a magnetic dipole moment; in other words the particle is a small electrically charged magnet with both a north pole and a south pole. When a magnet is inserted in a magnetic field the total energy is increased if the fields oppose as when north is next to north. The energy is decreased if the north pole of the magnet is nearer the south pole of the field. This energy difference is manifest when one turns the magnet around within the field. This is known to anyone who has played with magnets. This energy difference is proportional to the field strength. In a non-uniform field the magnet wants to move the direction that decreases the total energy.

The depicted magnetic poles produce a field which is stronger above near the wedge shaped pole. An electron with south pole up will be accelerated up as in the picture. Notice that the force on the particle does not depend on the particle’s charge or motion, only its magnetic dipole moment.

We now use either Newtonian or Einsteinian perspective and adopt a frame moving with the particle. Maxwells equations allow us to drop the electric field that was to counter the force on a charged article due to magnetic term in the Lorentz force.
F = e B×v
We guess that a non-uniform field would accelerate a spinning particle just as in the reported results and we assume that the particle is pushed either one of two equal but opposite forces with no intermediate cases.

Spin If you rotate an electron 360° you get a version that cancels a clone of the electron; the phase of the wave is reversed. If you send an electron thru two Spin filters oriented 180° apart the electron is blocked. See “Using the Spin Filter” here.

If you rotate a photon 180° about the axis parallel to its velocity you get the canceling photon; the phase of the wave is reversed. If you send a photon thru two polarizing filters oriented 90° apart, the photon is blocked.

If you rotate a gravity wave 90° about the axis parallel to its direction; the phase of the wave is reversed. If you sent a gravitational wave thru two ‘polarizing filters’ (such has not been designed) oriented at 45°, it would be blocked.

The ‘spins’ of these respective ‘particles’ are said to be 1/2, 1 and 2. I wish I knew the connection.