  For any Kac-Moody group $\mathbf{G}$, we prove that the Bruhat order on the
semidirect product of the Weyl group and the Tits cone for $\mathbf{G}$ is
strictly compatible with a $\mathbb{Z}$-valued length function. We conjecture
in general and prove for $\mathbf{G}$ of affine ADE type that the Bruhat order
is graded by this length function. We also formulate and discuss conjectures
relating the length function to intersections of "double-affine Schubert
varieties."
