Two eulerian trails of $K\_{2n+1}$ are defined to be equivalent if the orientations obtained by orienting the edges as traversed by the trails are isomorphic as digraph. How many non-equivalent trails are there? 



Non-equivalent eulerian trails in $K_{2n+1}$
If I understand your question, you are asking for the number of isomorphism classes of regular tournaments.


There are no exact formulas. See <http://oeis.org/A096368> for counts up to 15 vertices.


The asymptotic number is known. It is $RT(2n+1)/(2n+1)!$, where $RT(2n+1)$ is given in the abstract of [this paper](http://users.cecs.anu.edu.au/~bdm/papers/rt.pdf).


