  The eigenvalue spectra of staggered fermions with an Adams and/or Hoelbling
mass term are studied. The chiralities of the eigenmodes reflect whether the
chirality linked to the unflavored approximate (\gamma_5 \times 1) or the
flavored exact (\gamma_5 \times \xi_5) staggered symmetry is considered, and
which one of the RR, LR, RL, LL eigenmode definitions is used. In either case a
sensitivity to the topological charge of the gauge background is found. We
discuss how to remove the leading cut-off effects of these actions by means of
a properly tuned improvement term and/or the overlap procedure. The combination
of Symanzik improvement and link smearing radically improves the properties of
the physical branch.
