Which inverter do you have, specifically? A full generator/backup for the entire home?
There may be some downtime before it kicks in, hence the reboot.
There should be no more than 5% tolerance on modern ATX PSUs.
So +12V, for example, could see +11.40 V to +12.60 and be within spec.
BUT, focussing on the PSU; If you have a quality PSU and you experience power cuts often - then the PSU should be taking the brunt of any damage/surge/dirty power. I wouldn't expect it to last too long if it's doing that more than once in a while.
If you have a poor quality PSU, chances are good the PSU is protecting very little - and any damage (either occurring in small doses over time, or catastrophic damage in one event) will be felt by some/all of your components.
You should really look to a dedicated UPS for your system if power loss is a semi-common occurrence IMO.