\bar{\chi_{x}} \to S_{R}(R^{-1}x)\bar{\chi}_{R^{-1}x},\,\,\,\,
U_{\nu, x} \to U_{\nu, Rx}
\ ,
\label{rot1}\end{aligned}$$ where $R_{\rho\sigma}$ is the rotation $x_{\rho}\to x_{\sigma}$, $x_{\sigma}\to -x_{\rho}$, $x_{\tau}\to x_{\tau}$, $\tau \not= \rho, \sigma$ and $S_{R}(x)={1\over{2}}[1\pm\eta_{\rho}(x)\eta_{\sigma}(x)\mp\zeta_{\rho}(x)\zeta_{\sigma}(x)
+\eta_{\rho}(x)\eta_{\sigma}(x)\zeta_{\rho}(x)\zeta_{\sigma}(x)]$ with $\rho$$\sigma$. It is notable that the Adams-type fermion keeps this staggered rotation symmetry while the Hoelbling type loses it. The staggered charge conjugation transformation is given by $$\mathcal{C}:\,\,\chi_{x}\to\epsilon_{x}\bar{\chi}_{x}^{T},\,\,\,\,
\bar{\chi}_{x}\to-\epsilon_{x}\chi_{x}^{T},\,\,\,\,
U_{\nu,x} \to U_{\nu,x}^{*}
\ .
\label{C0}$$ The Adams-type fermion keeps this symmetry while the Hoelbling type loses it.

We next elucidate residual subgroups possessed by staggered-Wilson fermions, and discuss how to define physical discrete symmetries as Parity, Charge conjugation and Hypercubic symmetry. For this purpose we separate spin and flavor rotations in the above transformations. Here we utilize the momentum-space representation in [@GS; @DS]. In this representation we can define two set of clifford generators $\Gamma_{\mu}$ and $\Xi_{\mu}$, which operate on spinor and flavor spaces in the momentum-space field $\phi(p)$, respectively. (Details are shown in Appendix. \[SFS\].) Then the shift transformation translates into $$\mathcal{S}_{\mu}:\,\,\phi(p)\,\,\to\,\, \exp(ip_{\mu})\Xi_{\mu}\,\phi(p)
\ .
\label{shift2}$$ The axis reversal translates into $$\mathcal{I}_{\mu}:\phi(p)\,\,\to\,\,\Gamma_{\mu}\Gamma_{5}\Xi_{\mu}\Xi_{5}\,\phi(Ip)
\ .
\label{axis2}$$ The rotational transformation translates into $$\mathcal{R}_{\rho\sigma}:\,\,\phi(p)\,\,\to\,\,\exp({\pi\over{4}}\Gamma_{\rho}\Gamma_{\sigma})
\exp({\pi\over{4}}\Xi_{\rho}\Xi_{\sigma})\,\phi(R^{-1}p)
\ .
\label{rot2}$$ By using this representation we figure out residual discrete symmetries of staggered-Wilson fermions as 