They can and will. Intuitively and ethically, how you're describing is how it should work. It's how I thought it worked until I made a mistake like this.

After an hour on the phone with three different representatives, I finally got someone to explicitly confirm to me that leaving any interest accruing balance on a card will result in whatever your current balance is at the end of your next billing cycle all accruing interest, not just the remaining balance from last month.

If you leave $20 on your card after last billing cycle, then spend $500 this billing cycle, at the close of that billing cycle you will be charged a 1.66% on $520, even if all $500 of this billing cycle's charges were made 1 day before your statement closes.

Until you zero out your entire balance, the end of each billing cycle will charge interest on whatever the current balance is.

Yeah, its fucked up. Best case scenario is to learn the lesson here. Second best case is learning it with a $9 interest charge and an hour on the phone wondering how the hell $8.63 is 1.66% of $20.