> No, we cannot (completely) hear the shape of a drum, even if it is
> spinorial. Two metric fields with the same collection of eigenvalues 
> are called isospectral. There exist Dirac isospectral deformations;
> continuous 1-parameter families of mutually non- isometric metrics
> with the same Dirac spectrum have been constructed. They are of the
> form $M_s = G/F_s$, $s \in \mathbb{C}$, with $G$ a nilpotent group
> (e.g. the Heisenberg group) and $F_s$ a nilpotent subgroup. Also,
> there exist known examples of Laplace-isospectral 4-dimensional flat
> tori which are also Dirac-isospectral (at least for the trivial spin
> structure).

<A HREF="https://books.google.nl/books?id=Lobbf9sOd70C">Noncommutative Geometry and the Standard Model of Elementary Particle Physics</A> (page 304).