In some years, Chief Justice John G. Roberts Jr. does the honors. In others, it might be Justice Sonia Sotomayor or Justice Clarence Thomas presenting the squared-off hunks of marble affixed with the Supreme Court’s gilded seal.

Hewed from slabs left over from the 1930s construction of the nation’s high court and handed out in its magnificent Great Hall, they are a unique status symbol in a town that craves them. And while the ideological bents of the justices bestowing them might vary, there is one constant: All the recipients have given at least $5,000 to a charity favored by the justices, and, more often than not, the donors have a significant stake in the way the court decides cases.