Our Xantrex Prosine 2.0 Inverter watches the incoming voltage and line frequency and drops it out if out of spec.
It does not give protection for transients and voltage spikes. We've been lucky so far. (We pull the plug during lightning storms)
Forgot to mention that when the Xantrex cuts the power, its inverter function cuts in. The TV doesn't even flicker.
I do not know if all inverters work the same way but our main 120 for outlets and basic appliances comes through the inverter at all times so when we lose shore power there is no problem - and since lights are on 12 volt there is no problem there either. The problem can be with things like A/C and microwave and fridge - as well as potential harm to electronic devices. And if you run some things on incorrect voltage for a period of time they may all seem fine but will not last as long as they might on correct voltage. I am probably not the best at explaining all of this. But there is a good chance incorrect voltage will greatly reduce life expectancy of certain equipment. It is not just spikes but just continuous use at other than correct voltage that will cause early demise. A motor designed for 120 VAC but now running at 108 or 132 will wear out faster than if it ran at 120. There is also a difference between modified sine wave and true sine wave - many know this - some devices do not see 120 from a modified sine wave but they will see 120 from a true sine wave. A modified wave is clipped somewhat so a device may see it as only 108 volts - which could shorten life. I does get complicated.