  We analyze the decomposition of recently constructed unfaithful spinor
representations of K(E10) under its SO(9) x SO(9), and SO(9) x SO(2) subgroups,
respectively, where K(E10) is the `maximal compact' subgroup of the hyperbolic
Kac--Moody group E(10). We show that under these decompositions, respectively,
one and the same K(E10) spinor gives rise to both the fermionic fields of IIA
supergravity, and to the (chiral) fermionic fields of IIB supergravity. This
result is thus the fermionic analogue of the decomposition of E(10) under its
SO(9,9) and SL(9) x SL(2) subgroups, respectively, which yield the correct
bosonic multiplets of (massive) IIA and IIB supergravity. The essentially
unique Lagrangian for the supersymmetric E(10)/K(E10) sigma-model therefore can
also capture the dynamics of IIA and IIB including bosons and fermions in the
known truncations.
