Those positions boosted Trump electorally, particularly among aging whites in the industrial Midwest, where he secured his Electoral College victory. Flip-flopping on them would expose Trump to lines of attack that Democrats have employed successfully against Republican candidates in the past.
But traditional Republicans in Washington would very much like to overhaul those programs, and every time Trump adds one of those traditional Republicans to his cabinet, it raises the question of whether he'll weaken or abandon his safety-net populism.
This is the case with Rep. Tom Price (R-Ga.), Trump's pick to head the Department of Health and Human Services. Price is a longtime advocate of a Medicare plan that would convert the program from a defined benefit - where seniors are guaranteed certain health coverage - to a defined contribution, where the government would provide assistance for seniors to buy private health insurance plans.
Democrats call that plan "privatization," and they would love nothing more than for Trump to get behind it. They used Price's nomination on Tuesday to begin hammering Trump on that possibility.
"The nomination of Tom Price would put us on a direct path to end Medicare as we know it, which would raise health care costs and break a fundamental promise to seniors," Sen. Joe Donnelly (D-Ind.), who faces re-election in a red state in 2018, said in a press release announcing his opposition to Price.
Trump's release announcing his intent to nominate Price made no mention of any potential reforms to Medicare, privatization or otherwise. Since the election, though, House Speaker Paul Ryan and other GOP leaders - including Price - have raised the possibility of moving quickly to overhaul the program.
Such a move would be deeply unpopular, polls suggest. A 2015 Kaiser Family Foundation poll found only a quarter of respondents favor a shift to a Ryan/Price-style privatized Medicare system, compared to 70 percent who wanted Medicare to continue as is. The same survey found three-quarters of Medicare beneficiaries say the program works well.
Senior citizens are a group Trump can't afford to alienate. But Trump has signaled an openness to "modernizing" Medicare since his election, leaving deep confusion about whether he would actually keep his populist promise.
Until Trump himself rules out privatization - and nominees such as Price stand behind him - the question will linger.