  We only consider finite structures. With every totally ordered set $V$ and a
subset $P$ of $\binom{V}{2}$, we associate the underlying tournament ${\rm
Inv}(\underline{V}, P)$ obtained from the transitive tournament
$\underline{V}:=(V, \{(x,y) \in V \times V : x < y \})$ by reversing $P$, i.e.,
by reversing the arcs $(x,y)$ such that $\{x,y\} \in P$. The subset $P$ is a
pairing (of $\cup P$) if $|\cup P| = 2|P|$, a quasi-pairing (of $\cup P$) if
$|\cup P| = 2|P|-1$; it is irreducible if no nontrivial interval of $\cup P$ is
a union of connected components of the graph $(\cup P, P)$. In this paper, we
consider pairings and quasi-pairings in relation to tournaments. We establish
close relationships between irreducibility of pairings (or quasi-pairings) and
indecomposability of their underlying tournaments under modular decomposition.
For example, given a pairing $P$ of a totally ordered set $V$ of size at least
$6$, the pairing $P$ is irreducible if and only if the tournament ${\rm
Inv}(\underline{V}, P)$ is indecomposable. This is a consequence of a more
general result characterizing indecomposable tournaments obtained from
transitive tournaments by reversing pairings. We obtain analogous results in
the case of quasi-pairings.
