WASHINGTON — President-elect Donald J. Trump campaigned on an unusual mix of policy positions that pulled from both Republican and Democratic orthodoxy. He drew support from the mostly white working class of rural America that once leaned blue. And he said he would gladly work with the rival party to strike deals once he took office.

But with the top positions in his administration mostly filled out, it looks as if Mr. Trump could well pass on an early opportunity embraced by nearly every president since Franklin D. Roosevelt to reach across the aisle: appointing a member of the opposing party to his cabinet.

So far, of the 20 or so cabinet and other top-level advisory positions Mr. Trump has announced, not one has gone to a Democratic politician. With only a few seats left at the table, Democrats hope it will stay that way.