An article in Japanese gives more detail on the calculations for how the Koide formula is extended to the neutrinos. The various possible sign choices and how they work out with the charged leptons, neutrinos, and quarks are in a table on the bottom of page 23. A note on the coincidence with the Cabibbo angle is on the last page.
I presented some slides from this at the 2006 Joint Particle Physics meeting in Hawaii. The slides are here:
Actually, the above are the slides I was tried to present. I ended up with a slightly older version that had been downloaded earlier.
Meanwhile, I'm making steady progress in describing how one obtains the Koide formula, and more, from a modification to the foundations of quantum mechanics. I'm about three chapters away from the Koide mass formulas. I've been keeping the book on the web here: