  The $t$-onion star is the digraph obtained from a star with $2t$ leaves by
replacing every edge by a triple of arcs, where in $t$ triples we orient two
arcs away from the center, and in the remaining $t$ triples we orient two arcs
towards the center. Note that the $t$-onion star contains, as an immersion,
every digraph on $t$ vertices where each vertex has outdegree at most $2$ and
indegree at most $1$, or vice versa. We investigate the structure in digraphs
that exclude a fixed onion star as an immersion. The main discovery is that in
such digraphs, for some duality statements true in the undirected setting we
can prove their directed analogues. More specifically, we show the next two
statements.
  There is a function $f\colon \mathbb{N}\to \mathbb{N}$ satisfying the
following: If a digraph $D$ contains a set $X$ of $2t+1$ vertices such that for
any $x,y\in X$ there are $f(t)$ arc-disjoint paths from $x$ to $y$, then $D$
contains the $t$-onion star as an immersion.
  There is a function $g\colon \mathbb{N}\times \mathbb{N}\to \mathbb{N}$
satisfying the following: If $x$ and $y$ is a pair of vertices in a digraph $D$
such that there are at least $g(t,k)$ arc-disjoint paths from $x$ to $y$ and
there are at least $g(t,k)$ arc-disjoint paths from $y$ to $x$, then either $D$
contains the $t$-onion star as an immersion, or there is a family of $2k$
pairwise arc-disjoint paths with $k$ paths from $x$ to $y$ and $k$ paths from
$y$ to $x$.
