Symmetry for the spin highest weight module $V(Λ_n)$

For an affine Lie algebra $\mathfrak g$ of type $A^{(2)}_{2n}$ we get a special set of partitions for the highest weight modules $V(\Lambda)$, where $\Lambda=\Lambda_n$, and we also get a some results for the canonical basis elements. We demonstrate a symmetry of weights with fixed $n$-content around the weights gotten from the empty partition by adding multiples of the null root. We also find the highest power of the quantum parameter $q$ in an important set of canonical basis elements.