  We study the existence and instability properties of saddle-shaped solutions
of the semilinear elliptic equation $-\Delta u = f(u)$ in the whole $\R^{2m}$,
where $f$ is of bistable type. It is known that in dimension $2m=2$ there
exists a saddle-shaped solution. This is a solution which changes sign in
$\R^2$ and vanishes only on $\{|x_1|=|x_2|\}$. It is also known that this
solution is unstable.
  In this article we prove the existence of saddle-shaped solutions in every
even dimension, as well as their instability in the case of dimension $2m=4$.
More precisely, our main result establishes that if $2m=4$, every solution
vanishing on the Simons cone $\{(x^1,x^2)\in\R^m\times\R^m : |x^1|=|x^2|\}$ is
unstable outside of every compact set and, as a consequence, has infinite Morse
index. These results are relevant in connection with a conjecture of De Giorgi
extensively studied in recent years and for which the existence of a
counter-example in high dimensions is still an open problem.
