King Charles III quoted Oscar Wilde, joking that the British have everything in common with America “except, of course, language.” President Trump said the morning’s gloomy rain reminded him of a “beautiful British day” and noted that his mother thought young Prince Charles was “so cute.” Both men waxed poetic about the bonds between their countries.

And yet, on the first full day of a state visit focused on the shared history between the United States and Britain, the king sprinkled in some ever-so-subtle rebuttals to Mr. Trump. Charles spoke on Tuesday of the value of the trans-Atlantic alliance, the importance of checks and balances and his passion for the environment. He even spoke of his time in the Royal Navy, after Mr. Trump belittled British naval capabilities in recent weeks.

The king tucked his rejoinders into a mostly lighthearted speech to Congress on Tuesday afternoon and during evening remarks at a formal banquet at the White House.

“Please rest assured I am not here as part of some cunning rear-guard action!” the king told lawmakers in the afternoon, only the second time a British monarch had addressed Congress.

The mostly disciplined and careful public appearances by both Charles and Mr. Trump came at a dire moment in American-British relations, arguably at their lowest point in decades over the war in Iran and Mr. Trump’s scathing attacks on NATO.

But for a day (and maybe just a day), the special relationship that has developed over the past 250 years seemed — on the surface at least — special.

In a rarity for the Trump era, the president stuck mostly to his script during the day’s ceremonial events. He did not invite a horde of reporters into the Oval Office just before their meeting to field questions on Iran, the ballroom or Greenland in the presence of his visiting foreign dignitary. He did not lash out at another global ally.

In one apparently unscripted remark during the state dinner, Mr. Trump referred to the war in Iran and insisted that “Charles agrees with me.” It was an awkward moment because Charles studiously stays out of such matters of war and politics.

For the most part, though, Mr. Trump lavished the king with praise throughout the day.

“Before we ever proclaimed our independence, Americans carried within us the rarest of gifts — moral courage — and it came from a small but mighty kingdom from across the sea,” Mr. Trump said on Tuesday morning as he welcomed Charles to the White House.

There is little evidence in more recent history that an era of good feeling will last much beyond the departure of the royal couple’s jet from American shores on Thursday, particularly as Mr. Trump’s well-known affection for the royals does not extend to the British government.

Mr. Trump is furious at Britain for its refusal to join the fight against Iran, and his administration continues to accuse the British government of denying free speech to conservative voices. In London, Prime Minister Keir Starmer vows not to be dragged into another war of America’s choosing, and bristles at the president’s description of Britain’s aircraft carriers as nothing more than “toys.”

Those differences were never likely to be erased by the king’s first visit to the United States as the British monarch. By law and tradition, the king is supposed to rise above the disputes that often bedevil the leaders of both governments.

Mr. Trump was a guest of the royal family for a state dinner at Windsor Castle in September, an experience he described as “one of the highest honors of my life.” Months later, he belittled Mr. Starmer as a coward for not entering the U.S.-Israeli war with Iran.

“That was not very long ago and look where we are in terms of the bilateral relationship,” said Philippe Dickinson, deputy director at the Atlantic Council’s Transatlantic Security Initiative. “It can be cited as evidence by those who are going to make the case that it’s nice words one day and then forgotten the next day.”

Charles chose his words carefully during his public remarks.

During his speech to Congress, he appeared to address — obliquely — the Jeffrey Epstein scandal, which has caused political headaches for the Trump administration and led to a rupture in the royal family.

“In both of our countries, it is the very fact of our vibrant, diverse and free societies that gives us our collective strength, including to support victims of some of the ills that, so tragically, exist in both our societies today,” Charles said.

He also drew a standing ovation during his speech to Congress when he spoke about how the concept of checks and balances in American government has its roots in English history. Mr. Trump has worked to significantly expand executive power.

Charles said the U.S. Supreme Court Historical Society found that Magna Carta was cited in at least 160 Supreme Court cases since 1789, “not least as the foundation of the principle that executive power is subject to checks and balances.”

He spoke of “the natural wonders” of the United States and “our shared responsibility to safeguard nature, our most precious and irreplaceable asset.” Charles is an avid environmentalist; Mr. Trump, by contrast, pulled out of the Paris agreement on climate change, making the United States the only country in the world to abandon the international commitment to slow global warming.

The king spoke of his own service in the Royal Navy more than a half-century ago, and repeated Mr. Starmer’s assertion that Britain had “committed to the biggest sustained increase in defense spending since the Cold War.”

He also pushed back, gently, against Mr. Trump’s attacks on Britain and on the NATO alliance for not joining in the Iran war. After the Sept. 11, 2001, attacks, the king told lawmakers, “We answered the call together — as our people have done so for more than a century.”

And at the start of the evening’s state dinner, Charles recalled how the two nations have had “moments of difficulty” in the past, including in 1957 when Queen Elizabeth II visited the United States after the Suez Canal crisis.

“It is hard to imagine anything like that happening today,” Charles said, as some dinner guests laughed, causing Trump to turn and smile. “But it is not hard to see how important the relationship remains in matters both seen and unseen.”

While it was unclear whether the king’s appeal would be enough to mend the wounds in the trans-Atlantic relationship, Mr. Dickinson said the British were probably hoping the visit created a pathway to recovery.

“That’s why the government values the royal family as a diplomatic ace in the hand,” he said. “It’s not a magic wand, but it helps.”