  In this paper we study the su(m) spin Sutherland (trigonometric) model of D_N
type and its related spin chain of Haldane-Shastry type obtained by means of
Polychronakos's freezing trick. As in the rational case recently studied by the
authors, we show that these are new models, whose properties cannot be simply
deduced from those of their well-known BC_N counterparts by taking a suitable
limit. We identify the Weyl-invariant extended configuration space of the spin
dynamical model, which turns out to be the N-dimensional generalization of a
rhombic dodecahedron. This is in fact one of the reasons underlying the greater
complexity of the models studied in this paper in comparison with both their
rational and BC_N counterparts. By constructing a non-orthogonal basis of the
Hilbert space of the spin dynamical model on which its Hamiltonian acts
triangularly, we compute its spectrum in closed form. Using this result and
applying the freezing trick, we derive an exact expression for the partition
function of the associated Haldane-Shastry spin chain of D_N type.
