  We analyze the level-rank duality of untwisted and epsilon-twisted D-branes
of the so(N)_K WZW model. Untwisted D-branes of so(N)_K are characterized by
integrable tensor and spinor representations of so(N)_K. Level-rank duality
maps untwisted so(N)_K D-branes corresponding to (equivalence classes of)
tensor representations onto those of so(K)_N. The epsilon-twisted D-branes of
so(2n)_2k are characterized by (a subset of) integrable tensor and spinor
representations of so(2n-1)_2k+1. Level-rank duality maps spinor
epsilon-twisted so(2n)_2k D-branes onto those of so(2k)_2n. For both untwisted
and epsilon-twisted D-branes, we prove that the spectrum of an open string
ending on these D-branes is isomorphic to the spectrum of an open string ending
on the level-rank-dual D-branes.
