Twisted Hodge numbers
For a line bundle $\mathrm{L}$ the twisted Hodge numbers are
$$ \mathrm{h}^{p,q}(X,\mathrm{L}) = \dim \mathrm{H}^q(X,\Omega^p_X \otimes \mathrm{L}). $$For a general variety, letting $\mathrm{L}$ range over $\operatorname{Pic}(X)$ gives infinitely much (continuously varying) data. A hyperelliptic variety is special: its canonical bundle is torsion, $\omega_X^{\otimes m} \cong \mathcal{O}_X$ with $m = \operatorname{ord}(\omega_X)$, so the twists by powers of $\omega_X$ that matter for Hochschild theory repeat with period $m$:
$$ \mathrm{h}^{p,q}(X,\omega_X^{\otimes(j+m)}) = \mathrm{h}^{p,q}(X,\omega_X^{\otimes j}). $$There are therefore only $m$ distinct twists, a finite table of $(n+1)^2\cdot m$ integers, and each is a character multiplicity:
$$ \mathrm{h}^{p,q}(X,\omega_X^{\otimes j}) = \dim\big(\textstyle\bigwedge^p V^\vee \otimes \bigwedge^q\overline{V}^\vee \otimes \chi^{\,j}\big)^G = \frac{1}{\#G}\sum_{g\in G} e_p(\lambda(g)^{-1})\, e_q(\lambda(g))\, \chi(g)^j, $$where $\chi = \det\tilde\rho_X$ is the character of $\omega_X$ and $e_k$ the elementary symmetric polynomials in the eigenvalues $\lambda(g)$ of $\tilde\rho_X(g)$. The whole table is computed from the holonomy representation alone; it assembles the Hochschild–Serre cohomology and is a derived invariant.
The $m$ diamonds shown on each object page are exactly these twists, $\omega_X^{\otimes 0}, \dots, \omega_X^{\otimes(m-1)}$. They satisfy twisted Serre duality $\mathrm{h}^{p,q}(\omega^j) = \mathrm{h}^{n-p,n-q}(\omega^{-j})$, and $\mathrm{h}^{0,0}(\omega^j) = 1$ precisely when $\omega_X^{\otimes j} \cong \mathcal{O}_X$, which recovers $\operatorname{ord}(\omega_X)$.