  Let $M$ be a $2m$-dimensional compact Riemannian manifold. We show that the
spectrum of the Hodge Laplacian acting on $m$-forms does not determine whether
the manifold has boundary, nor does it determine the lengths of the closed
geodesics. Among the many examples are a projective space and a hemisphere that
have the same Hodge spectrum on 1-forms, and hyperbolic surfaces, mutually
isospectral on 1-forms, with different injectivity radii. The Hodge
$m$-spectrum also does not distinguish orbifolds from manifolds.
