  We study properties of two-color QCD at imaginary chemical potential ($\mu$)
from the viewpoint of the Roberge-Weiss (RW) periodicity, the charge
conjugation and the pseudo-reality. At $\mu=\pm i\pi T/2$, where $T$ is
temperature, the system is symmetric under the combination of the charge
conjugation ${\cal C}$ and the ${\mathbb Z}_{2}$ transformation. The symmetry,
called ${\cal C} {\mathbb Z}_{2}$ symmetry, is preserved at lower $T$ but
broken at higher $T$. The Polyakov-loop extended Nambu--Jona-Lasinio (PNJL)
model has the same properties as two-color QCD for ${\cal C} {\mathbb Z}_{2}$
symmetry and the pseudo-reality. The nontrivial correlation between the chiral
restoration and the deconfinement are investigated by introducing the
entanglement vertex in the PNJL model. The order of ${\cal C} {\mathbb Z}_{2}$
symmetry breaking at the RW endpoint is second-order when the correlation is
weak, but becomes first-order when the correlation is strong. We also
investigate the impact of the correlation on the phase diagram at real $\mu$.
