  We consider the elliptic equation $-\Delta u = f(u)$ in the whole $\R^{2m}$,
where $f$ is of bistable type. It is known that there exists a saddle-shaped
solution in $\R^{2m}$. This is a solution which changes sign in $\R^{2m}$ and
vanishes only on the Simons cone ${\mathcal C}=\{(x^1,x^2)\in\R^m\times\R^m:
|x^1|=|x^2|\}$. It is also known that these solutions are unstable in
dimensions 2 and 4.
  In this article we establish that when $2m=6$ every saddle-shaped solution is
unstable outside of every compact set and, as a consequence has infinite Morse
index. For this we establish the asymptotic behavior of saddle-shaped solutions
at infinity. Moreover we prove the existence of a minimal and a maximal
saddle-shaped solutions and derive monotonicity properties for the maximal
solution.
  These results are relevant in connection with a conjecture of De Giorgi on 1D
symmetry of certain solutions. Saddle-shaped solutions are the simplest
candidates, besides 1D solutions, to be global minimizers in high dimensions, a
property not yet established.
