  We consider the transformation reversing all arcs of a subset $X$ of the
vertex set of a tournament $T$. The \emph{index} of $T$, denoted by $i(T)$, is
the smallest number of subsets that must be reversed to make $T$ acyclic. It
turns out that critical tournaments and $(-1)$-critical tournaments can be
defined in terms of inversions (at most two for the former, at most four for
the latter). We interpret $i(T)$ as the minimum distance of $T$ to the
transitive tournaments on the same vertex set, and we interpret the distance
between two tournaments $T$ and $T'$ as the \emph{Boolean dimension} of a
graph, namely the Boolean sum of $T$ and $T'$. On $n$ vertices, the maximum
distance is at most $n-1$, whereas $i(n)$, the maximum of $i(T)$ over the
tournaments on $n$ vertices, satisfies $\frac {n-1}{2} - \log_{2}n \leq i(n)
\leq n-3$, for $n \geq 4$. Let $ \mathcal{I}_{m}^{< \omega}$ (resp.
$\mathcal{I}_{m}^{\leq \omega}$) be the class of finite (resp. at most
countable) tournaments $T$ such that $i(T) \leq m$. The class $\mathcal
{I}_{m}^{< \omega}$ is determined by finitely many obstructions. We give a
morphological description of the members of $\mathcal {I}_{1}^{< \omega}$ and a
description of the critical obstructions. We give an explicit description of an
universal tournament of the class $\mathcal{I}_{m}^{\leq \omega}$.
