Note: Do not do this yourself.
Have an electrician review the procedure and only do it if they think that it is a good idea and they can do it safely and correctly.
Here is a test you can have an electrician do.
1) Turn off all circuit breakers except one 120 volt breaker.
2) Measure the voltage from the breaker hot leg to neutral. Keep the volt meter connected.
2) Put a 120 volt load directly on that circuit breaker hot to neutral. The higher the better without overloading the breaker. Measure the current and the voltage
3) The voltage should remain relatively constant.
4) Measure for any current going through the ground wire.
5) Disconnect the ground from the panel and measure for voltage between the ground and the neutral. This is dangerous, that’s why an electrician needs to do it.
6) Watch the voltage and current to see if they change.
Basically, you’re load testing the neutral from the panel to the transformer to see if it can support a load without dropping voltage. If the voltage isn't right with no load, there's a problem. If the voltage drops a lot under load, there's a problem. If the voltage or current drop when the main ground is disconnected, there's a problem. If you have current going through the ground wire while it's connected, there's a problem.
When you have both hot legs with 120 volt loads and no neutral from the breaker panel to the transformer, the 120 volt loads are connected in series by the neutral wires from each load to the same neutral bus bar. So, the two hot legs carry the current.
However, if one hot leg is disconnected, the current can only return through the neutral to the transformer or through the neutral to the ground in the panel to the ground rod to the transformer ground rod to the transformer neutral.
BY doing the test, you can see if the neutral is disconnected or compromised.
Note: Do not do this yourself. Have an electrician review the procedure and only do it if they think that it is a good idea and they can do it safely and correctly.