  Symmetries and conservation laws of energy, linear momentum and angular
momentum play a central role in physics, in particular in nonlinear optics.
Recently, light fields with non trivial topology, such as polarization M\"obius
strips or torus-knot beams, have been unveiled. They cannot be associated to
well-defined values of orbital and spin angular momenta (OAM and SAM), but are
invariant under coordinated rotations, i.e. rotational symmetries that are
generated by the generalized angular momentum (GAM) operator, a mixture of the
OAM and SAM operators. The discovery of the GAM, which at variance with
integer-valued OAM and SAM, can have arbitrary value, and raises the question
of its conservation in nonlinear optical processes. By driving high harmonic
generation with a polarization M\"obius strip and implementing novel OAM
characterization methods in the XUV range, we experimentally observe the
conservation of the GAM, each harmonic carrying a precise half-integer GAM
charge equal to that of the fundamental field multiplied by the harmonic order.
The GAM is thus revealed as the appropriate quantum number to describe
nonlinear processes driven by light fields containing topological polarization
singularities.
