  In Part VII, we proved that the range of the big J-function in
permutation-equivariant genus-0 quantum K-theory is an overruled cone, and gave
its adelic characterization. Here we show that the ruling spaces are
D_q-modules in Novikov's variables, and moreover, that the whole cone is
invariant under a large group of symmetries defined in terms of q-difference
operators. We employ this for the explicit reconstruction of the cone from one
point on it, and apply the result to toric target spaces, when such a point is
given by the q-hypergeometric function.
