Before he began his remarks on health care policy at a White House event earlier this summer, Health Secretary Robert F. Kennedy Jr. first felt the need to praise one of President Trump’s passion projects.

Mr. Kennedy said that Mr. Trump’s new Oval Office décor had “transformed” a White House that comparatively looked “drab” when his uncle, former President John F. Kennedy, filled it. “Under your stewardship, it looks extraordinary today,” Mr. Kennedy said as Mr. Trump nodded in approval. “So thank you, Mr. President.”

During a three-hour cabinet meeting this week, it was Mr. Trump’s turn to support Mr. Kennedy’s endeavors: researching any link between vaccines and autism, a theory that many medical professionals and studies have debunked. “I know you’re looking very strongly at different things, and I hope you can come out with that as soon as possible,” Mr. Trump said.

The two exchanges crystallized the mutually beneficial nature of the relationship between the two men, an alliance driven by political calculation and a striking alignment of some of their views. Their ties have empowered Mr. Kennedy to push the boundaries of public health and science with the support of the president, who in turn benefits from Mr. Kennedy’s coalition of followers opposed to vaccines and health mandates.

It helps that Mr. Trump has long been dazzled by the storied nature of the Kennedy name.

“As someone who grew up during the Kennedy Camelot era, the idea of having a Kennedy report to him makes him feel good,” said Chris Meekins, a deputy assistant secretary for pandemic preparedness in the first Trump administration. “R.F.K. will be given a long-enough leash — will be given enough discretion — until there is a point when the political damage he’s doing to Republican efforts is greater than the political benefit he’s bringing.”