  Within the framework of residual symmetry, two $\mathbb{Z}_2$ type associate
$\mu\tau$ interchange symmetries robustly constrain the Dirac CP phase $\delta$
in a model independent way. Both of them predict simultaneous maximality of
$\delta$ and the atmospheric mixing angle $\theta_{23}$. We show how these well
known correlations will be changed if we generalize the $\mu\tau$ interchange
symmetry to a $\mu\tau$ mixing symmetry. In particular, we show that the
stringent condition of simultaneous maximality could be relaxed even with a
very small departure from the exact $\mu\tau$ interchange. In addition, the
present neutrino data on $\delta$ and $\theta_{23}$ can be explained better by
the mixing symmetry. After discussing the impact of the $\mu\tau$ mixing in
some realistic neutrino mass models, we show how the proposed mixing could be
realized with two simultaneous CP transformations which also lead to novel and
testable correlations between $\delta$ and the mixing angles $\theta_{ij}$.
Next we discuss in particular, the `three flavour regime' of leptogenesis
within the CP extended framework and show, unlike the ordinary CP extended
$\mu\tau$ interchange symmetry, a resonant leptogenesis is possible due the
generalization of $\mu\tau$ interchange to the $\mu\tau$ mixing and the
resulting baryon asymmetry always requires a nonmaximal $\theta_{23}$ owing to
the fact that the baryon to photon ratio $\eta_B$ vanishes in the exact limit
of $\theta_{23}=\pi/4$. This is one of the robust predictions of this
framework. The CP extended $\mu\tau$ mixing is also a novel example of a low
energy effective model that provides an important insight to the off-diagonal
terms of the flavour coupling matrix which have usually been neglected in
literature to compute the final baryon asymmetry, in particular in the models
with flavour symmetries.
