Floating Neutral
Out here on LI we experience them from time to time. I agree with the other posts that the ground rod will do virtually nothing, we have driven several down on top of each other, and get no ground. On LI we have water pipe grounds and have found that this will almost mirror the utility neutral. When we have problems as you describe if both the utility neutral and water main ground are open, we observed the differences in voltage you describe, hi and low on opposite legs, stand and watch the lights dim and increase in brightness, sometimes scary. Our utility doesn't compensate for much, and are protected in the tariff, however, they do pay for damages due to neutral being opened. Electronic type devices at 120 volts such as, microwave, TV, heating control boards etc will burn up pretty quickly. May want to reach out to some neighboring utilities and see what their policy is, if they will release that, and take it back to yours. I also thing the suggestion of homeowners is a good choice to investigate. One last thought, we have experienced high electric bills due to this condition, and actually tested it in a shop environment as well as houses, and confirmed the higher bills, which went against the engineers theory.