The physical (classical) significance of the spinor representation of an electron

Can we assign a physical (classical / correspondence principle) interpretation to the double rotation (720 degrees) required to describe electrons?

For an isolated electron, an overall 360 degree rotation gives a state that is physically identical to the original. It changes the sign of the quantum state-vector, $|\psi\rangle\rightarrow -|\psi\rangle$, but the overall complex coefficient of a quantum state-vector has no physical significance: the vector $z|\psi\rangle$ represents the same physical state for all non-zero complex numbers $z$. But, as mentioned in Andrew Steane's answer, things can become more interesting when more complex situations are considered. I'll say more about this below.

It is possibly simply a means of classification as a fermion, as opposed to other particles...

The spin-statistics theorem from relativistic QFT says that there is a connection between fermions and half-integer spin (and between bosons and integer spin). In nonrelativistic quantum mechanics, that theorem does not hold; we only enforce the connection manually because we know that nonrelativistic QM is supposed to be an approximation to relativistic QFT. So maybe the double-rotation property could be regarded as a means of classifying particles as fermions in relativistic QFT, but not in nonrelativistic quantum mechanics — again as mentioned in Andrew Steane's answer.

Is there any classical effect that we can point to, and say, "Without treating the electron as a spinor, i.e. needing a double rotation, (mathematically speaking), we could not explain this or that particular phenomenon"?

That depends on what is meant by "classical effect". There are innumerable effects with macroscopic consequences that can't be explained without treating electrons as spin-1/2 particles, such as ferromagnetism (as emphasized in my2cts's answer) and the specific ways that atoms and molecules interact with light, like the difference between fluorescence and phosphorescence. However, if the focus of the question is on the double rotation property of spin-1/2 particles, then one of the most direct manifestations is in the neutron interference experiments reviewed in this paper:

• "Theoretical and conceptual analysis of the celebrated $4\pi$-symmetry neutron interferometry experiments", https://arxiv.org/abs/1601.07053.

These are basically two-slit experiments with a macroscopic distance between the two paths in the interferometer. Diffraction in a crystal was used as a substitute for "slits." Magnets were arranged in a way that would cause precession of any neutron that passes through one of the paths, and the effect on the resulting two-slit interference pattern displays the effect of the sign-change under 360-degree rotations that characterizes spin-1/2 particles.

Despite the macroscopic distance between the slits, calling this a classical effect might be a stretch, because it relies on quantum interference between the two paths through the interferometer. If $|A\rangle$ and $|B\rangle$ represent the states of a neutron passing through path $A$ or path $B$ in the interferometer, then the experiment described above amounts to preparing the neutron in a state $|A\rangle+|B\rangle$, then applying a 360-degree rotation in the $B$-path to get the state $|A\rangle-|B\rangle$. Although an overall sign-change has no observable consequences, this relative sign-change does have physical consequences — which are observed in the resulting interference pattern in these experiments.

Whether or not this macroscopic effect deserves to be called classical is left to the discretion of the OP.

We cannot explain ferromagnetism without spin, nor any other form of permanent magnetism. Of course, you could argue that any phenomenon that requires spin is quantum mechanical.

This is a good question. I am not going to answer it fully, but I will offer some relevant pointers.

Spin is quantum mechanical in the sense that everything is, including orbital angular momentum, but it's not true to say that there is no classical (i.e. not quantum) mathematical apparatus for spin. You can have a classical particle possessing rest mass and spin, described by a spinor, and you can have a classical spinor field. In this sense, spin is no more quantum mechanical than any other physical property. It's just that historically physicists became aware of it at the same as they were discovering quantum mechanics, so they incorporated it into quantum mechanics from the start. Having done that, we can go back and incorporate it into classical mechanics if we want.

Having said that, I think one might be able to make a case that you can't have a relativistic quantum theory without spin, whereas you can have a relativistic classical theory without spin.

The double-rotation property can be related to something called 'angle doubling' in the correspondence between two groups called SU(2) and SO(3). I wrote an introduction (at senior undergraduate level) at https://arxiv.org/abs/1312.3824. (There may be other equally good or better introductions of course). For a classical observation that directly reveals the double-rotation property, I have not got a concrete suggestion, but I strongly suspect you would need a situation where one particle (say an electron) is interacting with another, and you rotate the first particle, for example using a magnetic field.