  We prove a pair of transformations relating elliptic hypergeometric integrals
of different dimensions, corresponding to the root systems BC_n and A_n; as a
special case, we recover some integral identities conjectured by van Diejen and
Spiridonov. For BC_n, we also consider their "Type II" integral. Their proof of
that integral, together with our transformation, gives rise to pairs of adjoint
integral operators; a different proof gives rise to pairs of adjoint difference
operators. These allow us to construct a family of biorthogonal abelian
functions generalizing the Koornwinder polynomials, and satisfying the
analogues of the Macdonald conjectures. Finally, we discuss some
transformations of Type II-style integrals. In particular, we find that adding
two parameters to the Type II integral gives an integral invariant under an
appropriate action of the Weyl group E_7.
