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The combination of $\nu_\mu$-$\nu_\tau$ exchange together with CP conjugationin the neutrino sector (known as $\mathrm{CP}^{\mu\tau}$ symmetry or$\mu\tau$-reflection) is known to predict the viable pattern:$\theta_{23}=45^\circ$, maximal Dirac CP phase and trivial Majorana phases. Weimplement such a CP symmetry as a new CP symmetry in theories with $A_4$flavor. The implementation in a complete renormalizable model leads to a newform for the neutrino mass matrix that leads to further predictions: normalhierarchical spectrum with lightest mass and $m_{\beta\beta}$($0\nu\beta\beta$) of only few meV, and either $\nu_1$ or $\nu_2$ has oppositeCP parity. An approximate $L_\mu-L_\tau$ symmetry arises naturally and controlsthe flavor structure of the model. The light neutrino masses are generated bythe extended seesaw mechanism with 6 right-handed neutrinos (RHNs). Therequirement of negligible one-loop corrections to light neutrino masses,validity of the extended seesaw approximation and not too long-lived BSM statesto comply with BBN essentially restricts the parameters of the model to a smallregion: three relatively light right-handed neutrinos at the GeV scale, heavierneutrinos at the electroweak scale and Yukawa couplings smaller than theelectron Yukawa. Such a small Yukawa couplings render these RHNs unobservablein terrestrial experiments.

A new and trivial CP symmetry for extended $A_4$ flavor