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Yep! Case in point: my team lead in a certain department truly disliked me, shouted at me "**** OFF YOU *****!" (for no logical reason-- she's legit crazy), causing me to get taken out of that department to prevent further issues. Well, that was a pg 13, and I was then moved to a position which was a pg 5, and I lost $0.50/hour, despite that not being my fault or issue at all. So yes, they will cut your pay if you move to a lower paygrade, lol.
My understanding from my days as HRTL is that it all depends on the ETL-HR. Some will lower the pay and some won't bother. I do know first hand though that if there is an issue which involves an HR to have to some investigating because of disputes and whatnot, if you voluntarily demote yourself you take the hit in pay versus if you are involuntarily demoted, you generally keep the pay. And even this is subjective HR to HR