The most telling remark in an evening of Republican triumphs Tuesday wasn’t Kentucky Republican Rand Paul’s declaration that America had produced a “tea party tidal wave,” though, in a week of watery metaphors, there was plenty of truth to that. The comment that captured the historic moment came from the Virginia Republican who in nine weeks will become the House majority whip, Eric Cantor, who spoke of a “second chance.”

The second political tidal wave of the 21st century — the first occurred but two years ago, when a swooning nation swept Barack Obama into the White House — produced second chances all around.

It gave the Republicans, consigned to obloquy if not oblivion in 2008, a chance to regain the status they covet above all others: the mantle of being the natural party of governance now that the New Deal has been shattered and is referenced more in textbooks than in text messages. Of all the transformations wrought this week, this was the most remarkable, for if there ever were a party that looked as if it were settling in for a long sojourn in the wilderness, it was the party of George W. Bush — who began a comeback of his own this week as details were released from his new memoir.

But it also gave President Obama a second chance. The need for this seemed inconceivable only two Novembers ago, when he was sovereign of all he surveyed, a pioneering figure headed toward history and a heroic battle to overhaul the American health care system, which accounted for one-seventh of the economy. It turns out that many Americans resented the way he tinkered with health care and objected to his stewardship of the other six-sevenths of the economy.

Yet this president, like the two before him who rebounded from mortifying midterm losses — Ronald Reagan (loss of 26 House seats in 1982, winner of 49 states in 1984) and Bill Clinton (loss of 52 House seats in 1994, winner of 31 states in 1996) — truly needs a second start.

Obama was the candidate whose intuitive sense of the American people’s yearnings sent him to the White House, yet he is the president who seems unable to hear American voices from the Oval Office. He was the candidate who promised permanent change in the capital, yet he is the president who is struggling this week to accommodate himself to the change his rivals will bring to Washington. He was the candidate who spoke so often and so eloquently of bipartisanship, yet he is the president who passed his signature pieces of legislation with almost no consultation with his Republican rivals and almost no votes from them either.

The need for a second start may be unwelcome in the West Wing, but so is an arrest for drunken driving. It is startling, it is humiliating, but it can serve as — here a fraught phrase from Jefferson comes to mind — a fire bell in the night. It tells everyone that change is needed — dramatic, open-minded change, in thought and action, in tone and timbre.

Every presidential rebuke is unhappy in its own way. Clinton saw his party relinquish control of both houses of Congress for the first time since he was 8 years old, while Reagan faced a lower approval rating and a higher unemployment rate than Obama. But the sheer number of House seats lost under Obama’s watch is staggering — more than in the Clinton debacle, twice as many as in the Reagan rebuke.

And that is by way of looking only at the numbers on the surface. Look below the surface, and the president’s challenge — no, more precisely: the president’s job — is formidable.

Leave aside for a moment the size and amplitude of the voters’ verdict and look at the president’s (dwindling) allies. Three of four Democratic voters this week said they were worried about the direction of the economy over the next year, according to surveys taken at polling stations for the National Election Pool. This in a country that overwhelmingly believes the economy is the biggest issue of the time.

Now let’s look at how bifurcated is the electorate. On the economy, 84 percent of Democratic voters said they thought Obama’s policies would help the country in the long run. A larger slice of Republican voters (88 percent) said they thought those policies would hurt the economy.

There are lots of battles ahead, some inside the Obama camp (where the recriminations will be more bitter than the rebuke), some between Democrats and Republicans in the House (where partisan resentment grows faster and more dangerously than bacteria in a petri dish), some between the Senate and the House (which march to their own drummers, the Senate’s always a few beats behind), some between governors (who love to trumpet their financial rectitude) and Washington (where financial rectitude is a contradiction in terms), and some in the new Republican coalition (where the regulars prefer decaf and the rebels are caffeinated, with no sugar).

One might have expected the conflict between Republican Party grandees and tea party insurgents to have been muted in the moment of victory, but by 5:03 a.m. Wednesday — lucky for the original tea partiers in 1773 Boston that there were no time stamps on their handbills and tracts — the war was on.

“Winning Republican control of the Senate at the expense of the platform and principles of the Republican Party is no victory,” said a statement e-mailed at that hour by the Tea Party Express. “Our nation got itself into the mess we face because Republicans sold out their principles and joined Democrats in supporting the policies of tax-spend-bailout.”

All this is evidence that the political wars did not end last week. The political wars of the century’s second decade — and the second chances they offer — only began.