follows. We first consider parity. Both staggered-Wilson fermions are invariant under $$\mathcal{I}_{s}\mathcal{S}_{4}\sim
\exp(ip_{4})\Gamma_{1}\Gamma_{2}\Gamma_{3}\Gamma_{5}\,\phi(-{\bf p},p_{4})\sim
\exp(ip_{4})\Gamma_{4}\,\phi(-{\bf p},p_{4})
\ ,  
\label{parity}$$ with $\mathcal{I}_{s}\equiv \mathcal{I}_{1}\mathcal{I}_{2}\mathcal{I}_{3}$. This is essentially the continuum parity transformation [@GS]. In the continuum limit the phase factor disappears and it results in the continuum parity transformations $P : \psi(p)\to\gamma_{4}\psi(-{\bf p},p_{4})$ for the Dirac fermion. We thus conclude both staggered-Wilson fermions possess symmetry leading to physical parity symmetry $P$. We note the simple combination of $\mu$-shift and $\mu$-axis reversal (shifted-axis reversal:$\mathcal{S}_{\mu}\mathcal{I}_{\mu}$) is also a symmetry of both fermions.

We next consider physical charge conjugation. In the case of Adams fermion the staggered charge conjugation symmetry $\mathcal{C}$ in Eq. (\[C0\]) remains intact. Thus, physical charge conjugation for the two-flavor branch can be formed in a similar way to usual staggered fermions as shown in [@GS] ($C\sim\mathcal{C}\mathcal{S}_{2}\mathcal{S}_{4}
\mathcal{I}_{2}\mathcal{I}_{4}$). On the other hand, the Hoelbling type breaks $\mathcal{C}$. In this case, however, we can define another charge conjugation by combining $\mathcal{C}$ with rotation transformation as $$\mathcal{C}_{T} : \,\,\mathcal{R}_{21}\mathcal{R}_{13}^{2}\mathcal{C}
\ .
\label{RRRC}$$ The Hoelbling action is invariant under this transformation. By using this symmetry we can define physical charge conjugation $C$ for one-flavor branch as in the Adams type. We thus conclude that both staggered-Wilson 