  We study a second CP symmetry compatible with the $A_4$ flavor group, which
interchanges the representations $\mathbf{1}'$ and $\mathbf{1}"$. We analyze
the lepton mixing patterns arising from the $A_4$ and CP symmetry broken to
residual subgroups $Z_3$ and $Z_2\times CP$ in the charged lepton and neutrino
sectors respectively. One phenomenologically viable mixing pattern is found,
and it predicts maximal atmospheric mixing angle as well as maximal Dirac CP
phase, trivial Majorana phase and the correlation
$\sin^2\theta_{12}\cos^2\theta_{13}=1/3$. We construct a concrete model based
on the $A_4$ and CP symmetry, the above interesting mixing pattern is achieved,
the observed charged lepton mass hierarchy is reproduced, and the reactor
mixing angle $\theta_{13}$ is of the correct order.
