  The antiferromagnetic insulating cuprate Ba$_3$Cu$_2$O$_4$Cl$_2$ contains
folded CuO$_2$ chains with four magnetic copper ions ($S=1/2$) per unit cell.
An underlying multiorbital Hubbard model is formulated and the superexchange
theory is developed to derive an effective spin Hamiltonian for this cuprate.
The resulting spin Hamiltonian involves a Dzyaloshinskii-Moriya term and a more
weak symmetric anisotropic exchange term besides the isotropic exchange
interaction. The corresponding Dzyaloshinskii-Moriya vectors of each magnetic
Cu-Cu bond in the chain reveal a well defined spatial order. Both, the
superexchange theory and the complementary group theoretical consideration,
lead to the same conclusion on the character of this order. The analysis of the
ground-state magnetic properties of the derived model leads to the prediction
of an additional noncollinear modulation of the antiferromagnetic structure.
This weak antiferromagnetism is restricted to one of the Cu sublattices.
