Unlike the Berry phase, the orbital angular momentum (OAM) of magnons with two-dimensional wave vector k in band n is not gauge invariant for arbitrary phase λn(k) and so is not physically observable. However, by integrating the OAM over the orientation ϕ of wave vector k, we construct a gauge-invariant function Fn(k). Like Fn(k), the average OAM for magnon band n in a circle of radius k is also gauge invariant and can be directly observed. We demonstrate these results for a ferromagnet on a honeycomb lattice with Dzyalloshinskii-Moriya interactions between next-nearest neighbor spins. With wave vectors k restricted to the first Brillouin zone, the angular averaged OAM Fn(k) then has opposite signs for lower and upper bands n=1 and 2 for all k.