  We study a two-parameter family of quantum spin systems on the complete
graph, which is the most general model invariant under the complex orthogonal
group. In spin $S=\frac{1}{2}$ it is equivalent to the XXZ model, and in spin
$S=1$ to the bilinear-biquadratic Heisenberg model. The paper is motivated by
the work of Bj\"ornberg, whose model is invariant under the (larger) complex
general linear group. In spin $S=\frac{1}{2}$ and $S=1$ we give an explicit
formula for the free energy for all values of the two parameters, and for spin
$S>1$ for when one of the parameters is non-negative. This allows us to draw
phase diagrams, and determine critical temperatures. For spin $S=\frac{1}{2}$
and $S=1$, we give the left and right derivatives as the strength parameter of
a certain magnetisation term tends to zero, and we give a formula for a certain
total spin observable, and heuristics for the set of extremal Gibbs states in
several regions of the phase diagrams, in the style of a recent paper of
Bj\"ornberg, Fr\"ohlich and Ueltschi.
  The key technical tool is expressing the partition function in terms of the
irreducible characters of the symmetric group and the Brauer algebra. The
parameters considered include, and go beyond, those for which the systems have
probabilistic representations as interchange processes.
