I put my desk fan on top of a book, and noticed it was Neuenschwander's book about Noether's theorem. That got me thinking about the symmetries of the fan: while there is a periodic symmetry for each $1/3$rd turn of the fan, there is a simple continuous screw symmetry along the time axis affecting the $x$-$y$ plane. So shouldn't there be a conserved Noether charge in this system associated with this symmetry?
Any screw transformation can be decomposed into a translation and a rotation (either along the screw axis or by decomposing it into a translation and a couple). Thus any conserved quantity related to such a symmetry would be decomposable into a combination of momentum and angular momentum.
In your particular case, with the translation being only along the time axis, I do believe the conserved quantity would be entirely angular momentum.