  As the theory is subject to a section condition, coordinates in double field
theory do not represent physical points in an injective manner. We argue that a
physical point should be rather one-to-one identified with a `gauge orbit' in
the coordinate space. The diffeomorphism symmetry then implies an invariance
under arbitrary reparametrizations of the gauge orbits. Within this generalized
sense of diffeomorphism, we show that a recently proposed tensorial
transformation rule for finite coordinate transformations is actually (i)
consistent with the standard exponential map, and further (ii) compatible with
the full covariance of the `semi-covariant' derivatives and curvatures after
projectors are properly imposed.
