Turns out the twiddle factor pattern, which I had thought was complicated, is actually simple. It looked complicated only because rounding error was nudging the argument of the twiddles back and forth between +/- pi.

The graph shows arg(twiddle)/pi for each twiddle factor for the FFT and the CS-SCHT.

The CS-SCHT's factors are just the FFT's factors quantized to [1, i, -1].