Let $G$ be a digraph with vertex set $V$. A module of $G$ is a set $M$ of vertices indistinguishable by the vertices outside $M$. A module of $G$ distinct from $V$ is a proper module of $G$. The digraph $G$ is reducible if its vertex set $V$ is the union of two proper modules. A digraph $G^{\prime}$, defined on $V$, is $\{-1\}$-hypomorphic to $G$ whenever the subdigraphs of $G$ and $G^{'}$ induced by $V\setminus\{x\}$ are isomorphic, for every vertex $x$. The digraph $G$ is $\{-1\}$-reconstructible if it is isomorphic to each digraph it is $\{-1\}$-hypomorphic to it. In this paper, given a reducible digraph $G$ having more than $4$ vertices, we describe the digraphs that are $\{-1\}$-hypomorphic to $G$. Our description is based on the decomposition of Gallai. As an immediate consequence, we obtain that every reducible digraph having more than $4$ vertices is $\{-1\}$-reconstructible; which improves the $\{-1\}$-reconstruction of disconnected graphs obtained by P. J. Kelly in $1957$ and that of reducible tournaments obtained by F. Harary and E. Palmer in $1967$.