The rotation of a Kerr black hole splits its low-frequency spectrum in two, so it was so far unclear why the known highly-damped resonances show no splitting. We find the missing, split sector, with spin squasinormal modes approaching the total reflection frequencies ω(n ∈N) =-Ω?J-iκ(n -s), where Ω, κ and ?J are the horizon's angular velocity, surface gravity, and induced change in angular momentum. Surprisingly, the new sector is at least partly polar, and corresponds to reversible Jtransitions. Its fundamental branch converges quickly, possibly affecting gravitational wave signals. Asimple interpretation of the Carter constant of motion is proposed.
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