  We show that the globally nilpotent G-operators corresponding to the factors
of the linear differential operators annihilating the multifold integrals
$\chi^{(n)}$ of the magnetic susceptibility of the Ising model ($n \le 6$) are
homomorphic to their adjoint. This property of being self-adjoint up to
operator homomorphisms, is equivalent to the fact that their symmetric square,
or their exterior square, have rational solutions. The differential Galois
groups are in the special orthogonal, or symplectic, groups. This self-adjoint
(up to operator equivalence) property means that the factor operators we
already know to be Derived from Geometry, are special globally nilpotent
operators: they correspond to "Special Geometries".
  Beyond the small order factor operators (occurring in the linear differential
operators associated with $ \chi^{(5)}$ and $ \chi^{(6)}$), and, in particular,
those associated with modular forms, we focus on the quite large order-twelve
and order-23 operators. We show that the order-twelve operator has an exterior
square which annihilates a rational solution. Then, its differential Galois
group is in the symplectic group $ Sp(12, \mathbb{C})$. The order-23 operator
is shown to factorize in an order-two operator and an order-21 operator. The
symmetric square of this order-21 operator has a rational solution. Its
differential Galois group is, thus, in the orthogonal group
  $ SO(21, \mathbb{C})$.
