First shown in Oberlack (2001) and significantly extended to higher moments in Oberlack et al. (2022), we presently show that in near-wall turbulent shear flows, i.e. specifically for the near-wall log-region, streamwise velocity moment scaling laws up to an arbitrary order can be derived using symmetry theory. Recently, we have extended the theory to wall-normal and spanwise velocity moments, which do not contain a mean-velocity. The new theoretical results are fully consistent to those for the streamwise scaling laws in Oberlack et al. (2022), and the symmetry predicts that the key scaling exponents ω in the log-region, are identical for all velocity moments. For its validation, and to compensate for the known weak convergence of the moment statistics of the spanwise and wall-normal velocities, we have again doubled the DNS run time of Hoyas et al. (2022) for wall shear Reynolds number of Re_τ=10⁴. DNS data and the extended symmetry theory show excellent agreement for both wall-normal and spanwise velocity moments in the above two areas, and, further, we confirm an essential result of the symmetry theory that the above scaling exponents are identical for all velocity moments. Further, the symmetry theory for arbitrary moments is detailed for all three velocities, and we explicitly show that the above scaling laws are invariant solutions of the infinite set of moment equations, i.e. a symmetry reduction has been achieved. In particular, the resulting equations contain the scaling parameter ω and this appears as a kind of eigenvalue of the reduced equations.