  The gravitational modes of the Kerr geometry include both quasinormal modes
and total-transmission modes. Sequences of these modes are parameterized by the
angular momentum of the black hole. The quasinormal and total-transmission
modes are usually distinct, having mode frequencies that are different at any
given value of the angular momentum. But a discrete and countably infinite
subset of the left-total-transmission modes are simultaneously quasinormal
modes. Most of these special modes exist along previously unknown branches of
the gravitational total-transmission modes. In this paper, we give detailed
plots of the total-transmission modes for harmonic indices $\ell=[2,7]$, with
special emphasis given to the $m=0$ modes which all contain previously unknown
branches. All of these unknown branches have purely imaginary mode frequencies.
We find that as we approach the Schwarzschild limit along these new branches,
the mode frequencies approach $-i\infty$ in stark contrast to the finite mode
frequency obtained in the Schwarzschild limit along the previously known
branches. We explain when and why, at certain frequencies, the
left-total-transmission modes are simultaneously quasinormal modes. At these
same frequencies, the right-total-transmission modes are missing. We also
derive analytic expressions for the asymptotic behavior of the
total-transmission mode frequencies, and for the values of the angular momentum
at which the modes are simultaneously quasinormal modes.
