  A distinguishing index of a (di)graph is the minimum number of colours in an
edge (or arc) colouring such that the identity is the only automorphism that
preserves that colouring. We investigate the minimum and maximum value of the
distinguishing index over all orientations of a given graph $G$. We present
sharp results for these parameters in terms of the distinguishing index of $G$
for trees, unbalanced bipartite graphs, traceable graphs and claw-free graphs.
With this, we answer the question of Meslem and Sopena.
