  This paper introduces two new spectral invariants of torsion-free
$\mathrm{G}_2$-structures on closed orbifolds and computes their values on all
Joyce orbifolds. These invariants are shown to be more discerning than the
$\overline{\nu}$-invariant of Crowley-Goette-Nordstr\"{o}m when applied to
Joyce orbifolds, and thus provide candidate tools for distinguishing between
Joyce manifolds. The invariants may be viewed as regularisations of the
classical Morse indices of the critical points of the Hitchin functionals on
closed and coclosed $\mathrm{G}_2$-structures respectively. In the case of
Joyce orbifolds, an interesting link with twisted Epstein $\zeta$-functions is
also observed.
