Invariant bilinear forms of odd dihedral doublets ($D_k\otimes D_k\supset 1$)

For the dihedral group of order $2m$ with odd $m$, let $D_k(r)=\mathrm{diag}({\zeta }^k,{\zeta }^{-k})$ and $D_k(s)=\left(\begin{array}{cc}0& 1\\ 1& 0\end{array}\right)$, where $\zeta =e^{2\pi i/m}$ and $1\le k\le (m-1)/2$. An invariant tensor defining a bilinear form satisfies $D_k(g{)}^{T}L{D}_k(g)=L$. The rotation forces $L_{11}=L_{22}=0$ and the reflection forces $L_{12}=L_{21}$. Thus the invariant tensor is a multiple of $\left(\begin{array}{cc}0& 1\\ 1& 0\end{array}\right)$, and a singlet Yukawa interaction gives two equal physical masses by singular value decomposition. Distinct inequivalent doublets admit no such bilinear form, by Schur lemma and their self-duality.