Discrete residual symmetries and flavour dependent CP symmetries consistentwith them have been used to constrain neutrino mixing angles and CP violatingphases. We discuss here role of such CP symmetries in obtaining a pseudo-Diracneutrino which can provide a pair of neutrinos responsible for the solarsplitting. It is shown that if (a) $3\times 3$ Majorana neutrino matrix $M_\nu$is invariant under a discrete $Z_2\times Z_2$ symmetry generated by $S_{1,2}$,(b) CP symmetry $X$ transform $M_\nu$ as $X^T M_\nu X=M_\nu^*$, and (c) $X$ and$S_{1,2}$ obey consistency conditions $X S_{1,2}^* X^\dagger=S_{2,1}$, then twoof the neutrino masses are degenerate independent of specific forms of $X$,$S_1$ and $S_2$. Explicit examples of this result are discussed in the contextof $\Delta(6 n^2)$ groups which can also be used to constrain neutrino mixingmatrix $U$. Degeneracy in two of the masses does not allow completedetermination of $U$ but it can also be fixed once the perturbations areintroduced. We consider explicit perturbations which break $Z_2\times Z_2$symmetries but respect CP. These are shown to remove the degeneracy and providea predictive description of neutrino spectrum. In particular, a correlation$\sin 2\theta_{23}\sin\delta_{CP}=\pm {\rm Im}[p]$ is obtained between theatmospheric mixing angle $\theta_{23}$ and the CP violating phase $\delta_{CP}$in terms of a group theoretically determined phase factor $p$. Experimentallyinteresting case $\theta_{23}=\frac{\pi}{4}$, $\delta_{CP}=\pm \frac{\pi}{2}$emerges for groups which predict purely imaginary $p$. We present detailedpredictions of the allowed ranges of neutrino mixing angles, phases and thelightest neutrino mass for three of the lowest $\Delta(6 n^2)$ groups with$n=2,4,6$.
