  We study a family of surfaces of general type that arises from the
intersections of two translates of the theta divisor on a principally polarized
complex abelian fourfold. In particular we determine the N\'eron-Severi
lattices of these surfaces and show that for a general abelian fourfold the
monodromy group of the associated variation of Hodge structures is a subgroup
of index at most two in the Weyl group W(E_6), whereas for a Jacobian variety
it degenerates to a subgroup of the symmetric group S_6.
