Mandel Ngan / AFP / Getty Images; Melina Mara / Washington Post

President Obama and House majority leader Eric Cantor engaged in a high-stakes test of wills during Wednesday’s debt-ceiling negotiations at the White House, trading dramatic ultimatums in the most intense round of talks yet. With tempers boiling over, Cantor took his grievances public in an unprecedented press conference after Obama issued a veto threat and told the Republican lawmaker he’d had “enough.”

The meeting began normally enough, with Obama welcoming eight congressional leaders from both parties to the White House. He made opening remarks and then called on Cantor. Cantor griped that the figure in cuts had been shrinking since last week. On July 7 — when Obama and House Speaker John Boehner proposed a grand bargain that Cantor helped bring down two days later in the face of a revolt from the right — the President offered $1.7 trillion in savings, Cantor said, as a baseline of agreement.

After the failure of the big $4.5 trillion deal, Cantor took over the negotiations for House Republicans. Suspiciously, he said, the baseline started shrinking. “When we were there yesterday, somehow the number became 1.6 to 1.7 to 1.8,” Cantor said he told the President. “So all of a sudden we are now drifting further downward, and today we now look to be below $1.4 trillion.” Democratic sources say the number, which came from the talks led by Biden — talks that collapsed when Cantor walked away from them two weeks ago — hasn’t changed. It has always been $1.5 trillion as a base with an additional $200 billion in savings that Republicans wanted and that the Administration had agreed to push for with congressional Democrats.

(MORE: Debt-Ceiling Negotiators’ Task: Getting Cantor to Deal)

But Cantor wasn’t done. Not only was the baseline number shrinking, he said, but the details had changed: the White House wanted $80 billion in Medicare spending and another $50 billion to fix the dual eligible problem in the Prescription Drug Program. “That’s something we never agreed to in the Biden talks,” Cantor said.

The President replied that though the White House still advocated for the $1.7 trillion figure, House and Senate Democrats could not support it, especially without revenue increases. The President added that the new conditions also came from congressional Democrats. “Maybe they ought to get it straight and see if they can get to $1.7 trillion,” Cantor told reporters in an unprecedented press conference outside of House votes in the Speaker’s Lobby after the White House meeting. Listen to it here:

Given that the two sides are so far apart — House Republicans have long demanded that the value of any increase to the federal borrowing limit be offset by deficit reductions, and it will take at least a $2.4 trillion hike to get through 2012 — Cantor offered to back off his insistence that there be only one debt-ceiling vote. (Some context: up until this point, House leadership aides had said the reason they were resistant to Senate minority leader Mitch McConnell’s suggestion of multiple votes is that they knew more than one vote would never pass their conference — i.e., they didn’t consider it a concession, but a necessity.) “And so I said, ‘Really, Mr. President, if you look at where we are right now, we are very far apart,’ ” Cantor told reporters. “And if you want the full $2.4 trillion increase and you won’t sign anything else, I don’t know if we can get there. And so I said I was willing to come off of my insistence that there be one vote that perhaps we could avoid default. That’s when he got very agitated.”

(MORE: Joe Klein Viewpoint: Cantor to the Woodshed)

Democratic sources coming out of the meeting allege that Cantor rudely interrupted the President three times — an accusation Cantor’s staff hotly disputes (“Eric waits to be recognized before speaking to the President,” says Cantor spokesman Brad Dayspring). Democratic sources say it was the third interruption that sparked the President’s temper.

The following paraphrased account of what Obama said next is cobbled together from Democratic and Republican sources:

What we’re seeing here confirms what the American people think is the worst about Washington: that everyone is more interested in posturing and political positioning and protecting their base than solving real problems. Eric, I could get well above the numbers the GOP is talking about with revenue increases. I am not afraid to veto this, and I will take that message and defend it to the American people. If we default, it will be a tax increase on every American. My responsibility is to the American people. I have reached the point where I say, ‘Enough.’ I have sat here long enough, and no other President — Ronald Reagan — wouldn’t sit here like this. I’ve reached my limit. We’ve reached the point where something’s got to give. You’ve either got to compromise on your dollar-for-dollar insistence or you compromise on the big deal, which means raising taxes. Eric, don’t call my bluff. I will go to the American people on this. This may bring my presidency down, but I will not yield on this.

According to Cantor, Obama then shoved back his chair and stormed out of the room. Democrats present at the meeting say there was no shoving or storming involved, but that he simply got up and said, “I’ll see you tomorrow.”

“I was somewhat taken aback because, you know, I was compromising,” Cantor told reporters. A Democratic source involved in the talks scoffs at Cantor’s “compromise.” “We’re not a banana republic,” the source says. “We’re not going to deal with this every three to six months. If you think it’s hard now, imagine how hard it’ll be in the middle of an election.”

The episode illustrates how far apart the two sides remain, even as the nation stands at the brink. But perhaps almost as troubling is Cantor’s litigation of the tension in the press. I have never seen negotiations broadcast so openly. It’s not a good sign. For every major successful bill I’ve covered on the Hill — Medicare Part D, the Bush tax cuts, the 2005 energy bill, CAFTA, the pension overhaul, TARP, the stimulus and health care reform — the principals always came out of the room and said, “We’re making progress,” or “Nice try, but I’m not going to negotiate with you,” or even, “I’m not going to negotiate with myself.”

(MORE: Eric Cantor, the GOP’s Hard-Line Lieutenant, Sways Debt Talks)

An agreement on raising the debt ceiling will not come from winning a spin war. If talks collapse, both sides will be blamed, and whatever they’re saying now won’t matter much in the face of economic disaster. The only solution at this point is to bite the bullet and draft a deal everyone is unhappy with. And the more public the process is — both for Cantor and the President — the harder it will become to reach a deal behind closed doors. Don’t get me wrong: I like getting the story as much as the next reporter. And if something big happens, we usually find out. But when talks blow up, there’s a real risk: if negotiators can’t trust one another not to snipe in the press — and this goes for both Cantor and the President, who has given his fair share of press conferences during this time — how can they trust one another to join arms and enact something as painful as deficit reduction?