Thanks to everyone for the help.

I did a little more digging and found a short summary from
Trans Coil that I have included at the end here. It's a PR
piece but makes for a good summary.

Would anyone know if the phases affect each other?
Do you get the same result if the cores were physically separate?

Does this thing work with single phase?

Tanks,
DOC

TCI KLR series three phase AC line reactors are intended for use as input
filters for adjustable speed DC drives and as input or output filters for
AC-PWM variable frequency drives. Drive performance is significantly
improved, the drives input rectifier is protected from failure or damage,
and drive harmonic demands are tamed with the addition of a K-rated line
reactor. KLR line reactors act as interface buffers between solid state
power circuits and the line or the motor. (Not unlike the surge protector
for your desk-top PC). All drives, in any application, will benefit when
applied with KLR series line reactors.