  We study the "fermionic billiards", i.e. the chaotic dynamics of the
gravitino, that arise in the near-spacelike-singularity limit of
eleven-dimensional supergravity and of its dimensional truncations (notably
four-dimensional simple supergravity). By exploiting the gravity-coset
correspondence, we show that the billiard dynamics of the gravitino is
described by a `spin extension' of the E(10) Weyl group, namely as a product of
90 degree `vector-spinor rotations' along certain simple-root-related
generators of the maximal compact subalgebra K(E(10)) of the hyperbolic
Kac--Moody algebra E(10). The `super-billiard' that combines the bosonic and
fermionic billiards is found to have a remarkably simple structure, which
exhibits a striking analogy with a polarized photon propagating in the
ten-dimensional Lorentzian Weyl chamber of E(10).
