  Two different notions of approximate Birkhoff-James orthogonality in nor\-med
linear spaces have been introduced by Dragomir and Chmie\-li\'n\-ski. In the
present paper we consider a global and a local approximate symmetry of the
Birkhoff-James orthogonality related to each of the two definitions. We prove
that the considered orthogonality is approximately symmetric in the sense of
Dragomir in all finite-dimensional Banach spaces. For the other case, we prove
that for finite-dimensional polyhedral Banach spaces, the approximate symmetry
of the orthogonality is equivalent to some newly introduced geometric property.
Our investigations complement and extend the scope of some recent results on a
global approximate symmetry of the Birkhoff-James orthogonality.
