  We prove that isomorphism of tournaments of twin width at most $k$ can be
decided in time $k^{O(\log k)}n^{O(1)}$. This implies that the isomorphism
problem for classes of tournaments of bounded or moderately growing twin width
is in polynomial time. By comparison, there are classes of undirected graphs of
bounded twin width that are isomorphism complete, that is, the isomorphism
problem for the classes is as hard as the general graph isomorphism problem.
Twin width is a graph parameter that has been introduced only recently (Bonnet
et al., FOCS 2020), but has received a lot of attention in structural graph
theory since then. On directed graphs, it is functionally smaller than clique
width. We prove that on tournaments (but not on general directed graphs) it is
also functionally smaller than directed tree width (and thus, the same also
holds for cut width and directed path width). Hence, our result implies that
tournament isomorphism testing is also fixed-parameter tractable when
parameterized by any of these parameters.
  Our isomorphism algorithm heavily employs group-theoretic techniques. This
seems to be necessary: as a second main result, we show that the combinatorial
Weisfeiler-Leman algorithm does not decide isomorphism of tournaments of twin
width at most 35 if its dimension is $o(n)$. (Throughout this abstract, $n$ is
the order of the input graphs.)
