  We study the C$^*$-algebra of Wiener-Hopf operators $A_\Omega$ on a cone
$\Omega$ with polyhedral base $P$. As is known, a sequence of symbol maps may
be defined, and their kernels give a filtration by ideals of $A_\Omega$, with
liminary subquotients. One may define $K$-group valued 'index maps' between the
subquotients. These form the $E^1$ term of the Atiyah-Hirzebruch type spectral
sequence induced by the filtration. We show that this $E^1$ term may, as a
complex, be identified with the cellular complex of $P$, considered as CW
complex by taking convex faces as cells. It follows that $A_\Omega$ is
$KK$-contractible, and that $A_\Omega/\mathbb K$ and $S$ are $KK$-equivalent.
Moreover, the isomorphism class of $A_\Omega$ is a complete invariant for the
combinatorial type of $P$.
