  In this paper we study the connection between the spectral condition of an
Hermite subdivision operator and polynomial reproduction properties of the
associated subdivision scheme. While it is known that in general the spectral
condition does not imply the reproduction of polynomials, we here prove that a
special spectral condition (defined by shifted monomials) is actually
equivalent to the reproduction of polynomials. We further put into evidence
that the sum rule of order $\ell>d$ associated with an Hermite subdivision
operator of order $d$ does not imply that the spectral condition of order
$\ell$ is satisfied, while it is known that these two concepts are equivalent
in the case $\ell=d$.
