  Transmission eigenchannels and associated eigenvalues, that give a full
account of wave propagation in random media, have recently emerged as a major
theme in theoretical and applied optics. Here we demonstrate, both analytically
and numerically, that in quasi one-dimensional ($1$D) diffusive samples, their
behavior is governed mostly by the asymmetry in the reflections of the sample
edges rather than by the absolute values of the reflection coefficients
themselves. We show that there exists a threshold value of the asymmetry
parameter, below which high transmission eigenchannels exist, giving rise to a
singularity in the distribution of the transmission eigenvalues, $\rho({\cal
T}\rightarrow 1)\sim(1-{\cal T})^{-\frac{1}{2}}$. At the threshold, $\rho({\cal
T})$ exhibits critical statistics with a distinct singularity $\sim(1-{\cal
T})^{-\frac{1}{3}}$; above it the high transmission eigenchannels disappear and
$\rho({\cal T})$ vanishes for ${\cal T}$ exceeding a maximal transmission
eigenvalue. We show that such statistical behavior of the transmission
eigenvalues can be explained in terms of effective cavities (resonators),
analogous to those in which the states are trapped in $1$D strong Anderson
localization. In particular, the $\rho ( \mathcal{T}) $-transition can be
mapped onto the shuffling of the resonator with perfect transmittance from the
sample center to the edge with stronger reflection. We also find a similar
transition in the distribution of resonant transmittances in $1$D layered
samples. These results reveal a physical connection between high transmission
eigenchannels in diffusive systems and $1$D strong Anderson localization. They
open up a fresh opportunity for practically useful application: controlling the
transparency of opaque media by tuning their coupling to the environment.
