  Unlike the Berry phase, the orbital angular momentum (OAM) of magnons with
two-dimensional wavevector k in band n is not gauge invariant for arbitrary
phase lambda_n(k). However, by integrating the OAM over the orientation $\phi $
of wavevector k, we construct a gauge-invariant function F_n(k). Like F_n(k),
the average OAM for magnon band n in a circle of radius k is also gauge
invariant. We demonstrate these results for a ferromagnet on a honeycomb
lattice with Dzyalloshinskii-Moriya interactions between next-nearest neighbor
spins. With wavevectors k restricted to the first Brillouin zone, the angular
averaged OAM F_n(k) then has opposite signs for lower and upper bands n=1 and 2
for all k.
