If spin (or angular momentum generally) is quantised, then it should be possible to measure the distinct orientations in an experiment. The experiment is called the Stern-Gerlach experiment, after the experimentalist who set it up first and the theoretician who proposed it.
The experiment consists of an oven with a pinhole opening. Silver atoms are vapourised in the oven and emerge through the pinhole. Another pinhole downstream is used to collimate the beam, i.e. to remove all atoms which stray off the beam axis. The beam is fed through a magnet with an inhomogeneous magnetic field, i.e. one whose field lines diverge. The silver atoms then hit a detector. In the original experiment they were just collected on a plate which was subsequently photographically processed.
Silver atoms are used because silver has a closed spherical arrangement of electrons in the lower states plus one extra electron. Therefore silver is a hydrogen-like atom with a rather large core and a single electron. Since all the core electrons are paired, the total (electronic) spin of the atom is that of the single outermost electron. This trick is necessary because an electron beam would be deflected by far more in the magnetic field due to the effect of the magnetic field on the moving charge (Lorentz force).
The magnet provides the principal axis needed to observe any directional quantisation.
The fact that the magnetic field is inhomogeneous causes magnetic moments, which are collinear with the spin orientation, to drift either further into or out of the magnetic field. The force acting on a magnetic moment equals the product of the moment and the divergence of the field: .
Classically, one would expect that the orientation of the magnetic moments can have any value. The result on the detector would then be a broad peak in the middle of the detector, broadened by the different effect of the divergent field on the continuous distribution of orientations.
The observation, however, shows two maxima either side of the centre. This shows that there are two distinct states with opposite magnetic moment and, by inference, spin.
An interesting thought experiment involves a series of Stern-Gerlach magnets. Suppose the spin-down beam is blocked off while the spin-up beam is fed through another inhomogeneous magnet. Since the beam is polarised after the first magnet, the second magnet doesn't do anything to the beam (other than deflect it a little more).
This time we turn the second magnet by 90o around the beam axis, i.e. perpendicular to the first one. The x-up beam splits into a y-up and a y-down component. Either we now have an (x-up, y-up) and an (x-up, y-down) beam, or we have destroyed the x-alignment by doing the experiment along the y axis.
To check which it is, we can feed the y-down beam emerging from the y-magnet into a third inhomogeneous magnet orientated in the same way as the first. If this beam is simultaneously (x-up, y-up) then the third magnet won't do anything to the x-up alignment. If the second magnet has destroyed the x-alignment, then the third magnet will act on a mixture of x-up and x-down states and separate them just as the first magnet did. In fact, the latter is true: probing the spin (or angular momentum) component in one direction destroys any information we may have had on another component.
And that's that! Now, here's the revision sheet.