Their policies and political outlooks diverge at nearly every turn, but one of the most constant and caustic clashes between former Vice President Joseph R. Biden Jr. and Senator Bernie Sanders is over Social Security.

The dispute goes back months, when Mr. Sanders’s campaign began attacking Mr. Biden and arguing he had previously voted to cut Social Security, something Mr. Biden has vehemently denied.

On the debate stage on Sunday, the two candidates ignited over the issue again, as Mr. Sanders tried to pin down Mr. Biden on an argument he made years ago from the Senate floor:

Sanders: Let me ask you a question, Joe. You’re right here with me. Have you been on the floor of the Senate, you were in the Senate for a few years, time and time again talking about the necessity, with pride, about cutting Social Security, cutting Medicare, cutting veterans programs?

Biden: No.

Sanders: You never said that?

Biden: No.

Sanders: All right, America, go to the website right now, go to the YouTube right now.