it could not lead to a correct continuum theory, and we would need to tune parameters to restore Lorentz symmetry. Indeed the recent study on symmetries of staggered-Wilson fermions by Sharpe [@Steve] reports that recovery of Lorentz symmetry requires fine-tuning of coefficients in the gluonic sector in lattice QCD with Hoelbling fermion.

To summarize, Adams fermion possesses physical parity, charge conjugation and hypercubic symmetries while Hoelbling fermion loses hypercubic as shown in Table. \[Table:sym\]. It seems that Hoelbling fermion cannot be straightforwardly applied to lattice QCD while Adams type can. We note that both staggered-Wilson fermions have proper parity symmetry, and we can discuss spontaneous parity symmetry breaking. Moreover, we may find some symptom due to Lorentz symmetry breaking in Hoelbling fermion in the parity-phase structure. This is another motivation to study parity-phase structure in strong-coupling lattice QCD.

In the end of this section, we comment on special symmetries in staggered-Wilson fermions without mass shift. Hoelbling fermion without the mass shift ($\eta_{\mu}D_{\mu}+M_{H}$) possesses special charge conjugation symmetry ($\mathcal{C}_{T}': \chi\to\bar\chi,\,
\bar{\chi}\to\chi$). This topic is beyond the scope of this work, but we note that it can do good to two flavors in the central branch. Use of the central branch is 