Hodge star eigenvalues on graded differential forms ($\mathrm{Spec}(\ast )=\{\pm 1\}(n\text{ odd}),\{\pm 1,\pm i\}(n\text{ even}>0)$)

Use the complex-linear Hodge star operator on the direct sum of all form degrees. Its square on degree $r$ is $(-1{)}^{r(n-r)}$. In positive odd dimension this is always one; in positive even dimension its sign is positive on even-degree forms and negative on odd-degree forms. Complementary basis wedges provide both roots of each sign, so the displayed eigenvalues all occur. On a single degree the Hodge star operator is an endomorphism only in middle degree; on middle degree $m$ in dimension $2m$ its possible eigenvalues are $\pm 1$ for even $m$ and $\pm i$ for odd $m$.