I don't know if a power supply would be happy with 240 split-phase and what effect it would have on efficiently.
I think MikeC may do this for PSU tests at 240V. You may want to check the SPCR PSU test methodology and perhaps MikeC may have some comment also.
A 120V neutral is not so much of a concern for making the PSU work in the first place. In some medical IPS installations with a floating earth, neutral and live both sit at 120V, 180 degrees out of phase and this reliably works for powering computers. A concern I would raise from operational experience is to make sure that there is good earthing and all equipment is powered from this same supply. Different equipment sets up a ground connection quite differently and if you are not careful you might find an AC voltage flowing over the Ground wire of a video cable between your monitor and computer, each having a different floated ground voltage. You shouldn't be able to fry anything, but signal quality is at risk.