  This paper addresses saddle-shaped solutions to the semilinear equation $L_K
u = f(u)$ in $\mathbb{R}^{2m}$, where $L_K$ is a linear elliptic
integro-differential operator with a radially symmetric kernel $K$, and $f$ is
of Allen-Cahn type. Saddle-shaped solutions are doubly radial, odd with respect
to the Simons cone $\{(x', x'') \in \mathbb{R}^m \times \mathbb{R}^m \, : \,
|x'| = |x''|\}$, and vanish only in this set.
  We establish the uniqueness and the asymptotic behavior of the saddle-shaped
solution. For this, we prove a Liouville type result, the one-dimensional
symmetry of positive solutions to semilinear problems in a half-space, and
maximum principles in "narrow" sets. The existence of the solution was already
proved in part I of this work.
