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:
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.