Sorry for the misleading title, I actually mean the following:

The $n$-th hyperoctahedral group, also known as the Weyl group of $\mathrm{Sp}_{2n}$ and of $\mathrm{SO}_{2n+1}$, is isomorphic to the group of $n\times n$ signed permutation matrices.

On the other hand, viewing $\mathrm{O}_n$, the usual orthogonal group, as a (non-connected) reductive group scheme over $\mathbb{Z}$, we see that $\mathrm{O}_n(\mathbb{Z})$ is also the group of signed permutation matrices. So, surprisingly, the rational point subgroup of a reductive group becomes the Weyl group of another reductive group.

Is there a conceptual (reductive group theoretic or representation theoretic) explanation for this coincidence?