  We establish the uniqueness of a saddle-shaped solution to the diffusion
equation $-\Delta u = f(u)$ in all of $\mathbb{R}^{2m}$, where $f$ is of
bistable type, in every even dimension $2m \geq 2$. In addition, we prove its
stability whenever $2m \geq 14$. Saddle-shaped solutions are odd with respect
to the Simons cone ${\mathcal C} = \{(x^1,x^2) \in \mathbb{R}^m \times
\mathbb{R}^m : |x^1|=|x^2| \}$ and exist in all even dimensions. Their
uniqueness was only known when $2m=2$. On the other hand, they are known to be
unstable in dimensions 2, 4, and 6. Their stability in dimensions 8, 10, and 12
remains an open question. In addition, since the Simons cone minimizes area
when $2m \geq 8$, saddle-shaped solutions are expected to be global minimizers
when $2m \geq 8$, or at least in higher dimensions. This is a property stronger
than stability which is not yet established in any dimension.
