  We present a systematic study on minimally perturbed neutrino mass matrices
which at the leading order give rise to Tri-BiMaximal (TBM) mixing due to a
residual $\mathbb{Z}_2\times \mathbb{Z}_2^{\mu\tau}$ Klein symmetry in the
neutrino mass term of the low energy effective seesaw Lagrangian. Considering
only the breaking of $\mathbb{Z}_2^{\mu\tau}$ with two relevant breaking
parameters ($\epsilon_{4,6}^\prime$), after a comprehensive numerical analysis,
we show that the phenomenologically viable case in this scenario is a special
case of TM1 mixing. For this class of models, from the phenomenological
perspective, one always needs large breaking (more than $ 45\%$) in one of the
breaking parameters. However, to be consistent the maximal mixing of
$\theta_{23}$, while more than $ 35\%$ breaking is needed in the other, a range
$49.4^\circ-53^\circ$ and $38^\circ-40^\circ$ could be probed allowing breaking
up to $ 25\%$ in the same parameter. Thus though this model cannot distinguish
the octant of $\theta_{23}$, non-maximal mixing is preferred from the viewpoint
of small breaking. The model is also interesting from leptogenesis perspective.
Unlike the standard $N_1$-leptogenesis scenario, here all the RH neutrinos
contribute to lepton asymmetry due to the small mass splitting controlled by
the $\mathbb{Z}_2^{\mu\tau}$ breaking parameters. Inclusion of flavour coupling
effects (In general, which have been partially included in all the leptogenesis
studies in perturbed TBM framework) makes our analysis and results pertaining
to a successful leptogenesis more accurate than any other studies in existing
literature. Finally, in the context of recent discovery of the ultra high
energy (UHE) neutrino events at IceCube, assuming UHE neutrinos originate from
purely astrophysical sources, we obtain prediction on the neutrino flux ratios
at neutrino telescopes.
