Even the participants in President Biden and Donald J. Trump’s overlapping visits to Texas on Thursday seemed to sense there was something remarkable about their near encounter along the southern border.

Rarely do the current and former commanders in chief arrive on the same scene on the same day to present such sharply different approaches to an issue as intractable as immigration. Even rarer still was the reality that the two men are most likely hurtling toward a rematch in November.

“Today is a day of extraordinary contrast,” declared Gov. Greg Abbott of Texas, who had appeared alongside Mr. Trump.

But the dueling border events were about something even more fundamental than immigration policy. They spoke to the competing visions of power and presidency that are at stake in 2024 — of autocracy and the value of democracy itself.

Perhaps the most surprising facet of the split screen was that Mr. Trump and Mr. Biden agreed on some of the basic contours of the border problem: that the current situation, with migrant crossings setting a new monthly record of nearly 250,000 in December, is unsustainable.

“It’s long past time to act,” Mr. Biden said.