Tuesday’s elections, which produced a resounding Republican victory in New Jersey and a dispiriting loss for the GOP in Virginia, highlighted the challenges ahead for a badly divided party — and will probably intensify an internal debate about how to win back the White House in 2016.

At a time when the party’s image has sunk to record lows nationally, the results of the gubernatorial elections will reverberate far beyond the borders of Virginia and New Jersey. Off-year elections are hardly foolproof in predicting the future, but as GOP leaders digest what happened Tuesday, the lessons they take away from the races after their autumn of discontent will shape the coming rounds.

In New Jersey, Gov. Chris Christie (R) rolled to reelection by a margin that will make him a leading contender for his party’s presidential nomination in 2016, should he decide to run. His victory in a solidly blue state will be touted as a model for a party that needs to expand its coalition in national campaigns. But will the formula Christie employed in New Jersey work in Republican primaries and caucuses or in a national election for president?

In Virginia, Republican Attorney General Ken Cuccinelli II, a tea party favorite, came closer than many expected but fell short in his race against Democratic businessman and party fundraiser Terry McAuliffe. What Republicans will debate was whether Cuccinelli was personally too conservative — and his party too toxic after the recent government shutdown — for what is now a classic swing state.

The outcomes set up a battle for power between competing wings of the Republican Party. Call it the establishment vs. the tea party, or the gubernatorial wing against the congressional wing. This competition is less about ideology or policy — there is no disunity, for example, when it comes to the party’s dislike of President Obama’s Affordable Care Act — than about purity vs. pragmatism, tactics and strategy. Or, as Christie has put it, it is about winning an argument vs. winning elections.

See how the vote breaks down among groups and how much those groups have shifted since 2009. Hover over the bars to see the size of the voting group. Results for typical characteristics have a margin of error of four percentage points. Current results are preliminary and will change as voting returns come in. See bottom of page for exit poll methodology.

Christie’s campaign will embolden the establishment wing and many of the GOP’s major fundraisers, who have been on the defensive as tea party conservatives flexed their muscles in Republican primaries and in the battle in Congress that led to the shutdown. Those establishment forces have vowed to become more active in opposing the insurgency that has moved Republicans to the right.

But Cuccinelli’s narrow loss will not necessarily change the underlying shape of the party or the attitudes of many grass-roots conservatives about the need to oppose Obama and the Democrats at every turn. Cuccinelli ran hard against Obama’s health-care law in the closing days of the campaign, and many Republicans might conclude that with another week or two, he would have prevailed. They will make opposition to the health-care law the first page of the playbook for 2014 races, and possibly for 2016 as well.

Nor will Christie’s victory necessarily translate easily into a winning strategy in a national election. His win was personal, not an endorsement of his party. What has worked for him in New Jersey may or may not be easily transported to states with very different electorates. One exit poll question pitted Christie against Hillary Rodham Clinton in a hypothetical presidential race. Even in New Jersey, Clinton prevails.

What sank Cuccinelli will be the topic of debate among Republicans as they consider the tea party’s culpability in the defeat. There is little doubt that Cuccinelli’s past policies and statements badly hurt him. McAuliffe’s campaign, which had a sizable financial edge, pounded Cuccinelli early, leaving him deeply wounded politically.

But factors beyond his control also contributed. One was an ethics scandal that engulfed Gov. Robert F. McDonnell (R) and that touched Cuccinelli. The other was the shutdown, though Cuccinelli compounded the problem by inviting an architect of the GOP strategy, Sen. Ted Cruz (R-Tex.), to campaign for him.

Christie did not run from his party, noting throughout the campaign that he opposed abortion rights and same-sex marriage and favored the kind of tax cuts that are part of conservative orthodoxy. As one of many governors who criticized the shutdown, he also kept himself insulated from its damage.

His success was testament to his powerful personality, his authenticity and his governing strategy, which combined conservative principles with a willingness to work with Democrats. Most significantly, perhaps, was the leadership he showed after Hurricane Sandy devastated parts of his state a year ago.

Chris Christie, who won reelection on Tuesday and Ken Cuccinelli, who lost his race for Virginia governor, represent different sides of today's GOP. So where does the party go as it gears up for 2016? Richard Fowler and Sabrina Schaeffer discuss with PostTV’s On Background. (The Washington Post)

What Christie pointed to throughout his campaign was winning a bigger share of traditionally Democratic constituencies. Early exit polls showed that he boosted his numbers over his 2009 election among Hispanics and blacks.

One glaring contrast with Cuccinelli was the women’s vote. Christie was winning a majority of the votes among men and women. Cuccinelli was losing among women, and he was losing among unmarried women, a key Democratic constituency, by better than 2 to 1.

Mike Murphy, a GOP strategist who sides with the establishment wing of the party, said the shorthand from Tuesday’s results was plain. “Christie’s a how-to manual and Virginia is a how-not-to manual.”

John Brabender, the chief strategist for the presidential campaign of former senator Rick Santorum (R-Pa.), sought to throttle back those kinds of assessments. “I think you’ll see a lot of people try to read in that the moderate [Christie] did well and the conservative [Cuccinelli] struggled,” he said. “I think that’s a grand oversimplification.”

As they look to 2016, Republicans can anticipate a potentially brawling nomination contest. A GOP strategist who spoke on the condition of anonymity to be candid said many primary and caucus voters will not look favorably on Christie’s willingness to work with Democrats. “They want you pure,” he said.

That suggests that the debate about the way forward for the Republicans will continue to rage.