The $\mu$-$\tau$ reflection symmetry
$(\nu_{e},\thinspace\nu_{\mu},\thinspace\nu_{\tau})\rightarrow(\overline{\nu}_{e},\thinspace\overline{\nu}_{\tau},\thinspace\overline{\nu}_{\mu})$
and the TM1 mixing (a PMNS matrix with the first column fixed to the TBM form)
are both well compatible with experiments. If both approaches are
simultaneously assumed, all lepton mixing parameters except for $\theta_{13}$
are predicted. In particular, one expects maximal CP violation
($|\delta|=90^{\circ}$), maximal atmospheric mixing ($\theta_{23}=45^{\circ}$),
a slightly less-than-TBM solar mixing angle ($\theta_{12}\approx34^{\circ}$),
as well as values of $0$ or $\pi$ for the two Majorana phases. We study the
renormalization stability of this highly predictive framework when neutrino
mass is described by an effective Weinberg operator and by the type I seesaw
mechanism, both in the Standard Model and with supersymmetry.Comment: 12 pages, comments added, version to appear in PR