  We introduce the price of symmetrisation, a concept that aims to compare
fundamental differences (gap and quotient) between values of a given graph
invariant for digraphs and the values of the same invariant of the symmetric
versions of these digraphs. Basically, given some invariant our goal is to
characterise digraphs that maximise price of symmetrisation. In particular, we
show that for some invariants, as diameter or domination number, the problem is
easy.
  The main contribution of this paper is about (partial) results on the price
of symmetrisation of the average distance. It appears to be much more intricate
than the simple cases mentioned above. First, we state a conjecture about
digraphs that maximise this price of symmetrisation. Then, we prove that this
conjecture is true for some particular class of digraphs (called bags) but it
remains open for general digraphs. Moreover, we study several graph
transformations in order to remove some configurations that do not appear in
the conjectured extremal digraphs.
