In the classic movie “The Untouchables,” the street-smart cop Jim Malone explains to his golden-boy partner Eliot Ness that things will have to get dirty if they intend to bring down Al Capone: “You see what I’m saying is, what are you prepared to do?” That’s the question for the GOP as it sifts through the ashes of this week’s cliff deal.The tax-hike extravaganza that President Obama signed on Wednesday was Round One of a bigger deficit fight, and the GOP was battered badly. Poor messaging, an internal tax feud, and a miscalculation of the president’s tactics—all combined to land the private economy with a monstrous tax bill, and the Republican Party with a black eye.On to Round Two, which will center on the debt ceiling due to hit in February. Republicans are convinced they can win this one. Their thinking? The president can’t use the threat of higher middle-class taxes to force the GOP to yield. Without the middle class as a hostage in the negotiations, they believe, the debt-ceiling debate will be entirely on spending and Mr. Obama’s failure to confront the nation’s $16 trillion debt.

The White House feels this keenly, as exhibited by the ferocious threats the president leveled in the aftermath of his tax-increase victory. “If the Republicans think that I will finish the job of deficit reduction through spending cuts alone, that’s not how it’s going to work,” he said in a Monday press conference. Translation: He will demand more taxes.

The president is steeling himself for Round Two. Are Republicans? For all the happy talk about their leverage in the debt-ceiling fight, this is going to be dirty. What are they prepared to do? They face three big questions.

Question one: Do they mean it? In the abstract, the debt ceiling is a powerful tool for forcing the president to give in to spending cuts. The Obama Treasury can’t pay the bills without say-so from the Republican House, so the House holds all the cards.

In the non-abstract, failure to raise government borrowing limits means U.S. default—and with it potential credit downgrades, market panic and resulting economic distress. Is the GOP willing to inflict that on the economy? If Republican members instead run for cover, as they did with the cliff, the GOP will have been exposed as bluffers, and the administration will never again have to fear the debt ceiling. Republicans have to consider if they are willing to take that risk.

Question two: What do they want? Throughout the fiscal-cliff negotiations, the Republicans kept thinking Mr. Obama would sign on to entitlement reform, giving both parties political cover. In this vain hope, the GOP shrunk from laying out its specific demands on Medicare, Social Security and Medicaid.

President Obama didn’t bite, and he won’t in the future. The GOP must know by now that the president’s only goal is to water down any reform proposals. So their only chance of making a dent in the debt is to begin bold.

Do House Republicans have the courage to lay out big demands (say, premium support for Medicare or block grants for Medicaid), send a bill to the Senate, and sell entitlement reform to the public? If they can’t face the demagoguery that Democrats will use against them for making substantive proposals on entitlements, then they have already resigned themselves to piddling spending cuts that only nibble around the entitlement edges. Is that worth an epic showdown?

Question three: What other hostages are Republicans willing to see shot? Knowing he has lost his tax trump card, Mr. Obama seamlessly moved on this week to the defense budget. The cliff deal turns off the automatic sequester cuts to the military for only two months, and Mr. Obama intends to make further tax hikes the price for anything longer.

Are the GOP’s defense hawks willing to stomach those cuts as a price for entitlement reform? Having publicly campaigned against this slashing of the military, can the party stare down the president with a unified position? Mr. Obama is betting they can’t, which is precisely why he ensured in the cliff deal that the sequester kicks in at the very time of the debt-ceiling fight.

Only the GOP can answer these questions, but the point here is that Republicans had better have answered them—and clearly—before they step into the ring. The president has every intention of playing them exactly as he did in the cliff, and in 2011.

Mr. Obama will lay out tax-hike demands, give no quarter on spending, not waver and, as the deadline approaches, use his bully pulpit and the media to cow the GOP into the sort of wrangling that led to this week’s defeat.

If the Republican strategy isn’t crystal clear, if the party is again fractured, then Round Two is already Mr. Obama’s. So once again: What, exactly, is the GOP prepared to do?

Write to kim@wsj.com