  Following the recent studies of the trickiness in spin and orbital angular
momentum of the vector gauge fields, we perform here a parallel analysis for
the tensor gauge field, which has certain relation to gravitation. Similarly to
the vector case, we find a nice feature that after removing all gauge degrees
of freedom the angular momentum of the tensor gauge field vanishes for a
stationary system. This angular momentum also shows a one-parameter invariance
over the infinitely many ways of complete gauge fixing for the tensor field.
The tensor gauge coupling, however, does exhibit a critical difference from the
vector gauge coupling that it may induce intrinsic interaction terms into the
spatial translation and rotation generators, leaving none of the ten Poincar\'e
generators interaction-free.
