Democratic Rep.-elect Kathy Hochul’s victory in a New York special election on Tuesday highlights Republicans’ political problem on Medicare. (AP Photo/Harry Scull Jr/The Buffalo News)

On the one hand, former House Speaker Newt Gingrich, a 2012 presidential candidate, was castigated by conservatives for his criticism of Wisconsin Rep. Paul Ryan’s budget — in particular the changes in Medicare it proposed.

On the other, state Assemblywoman Jane Corwin (R) lost a reliably Republican seat in a special election on Tuesday thanks, in no small part, to her decision to embrace the Ryan plan.

The reality is that the total fealty to the Ryan budget increasingly demanded of Republican presidential candidates by the party’s base runs directly counter to the unpopularity of making drastic cuts to Medicare among the general electorate.

Not only is that a major political problem for the party but it’s one without a simple solution.

The reason Medicare presents such a conundrum for Republicans is that the constituencies the party’s presidential candidates are trying to reach run are not the same sort of folks who will help the GOP maintain its House majority or win control in the Senate in 2012.

People like Gingrich, former Utah governor Jon Huntsman Jr. — who has said he would have voted for the Ryan plan — and former Massachusetts governor Mitt Romney are doing everything they can to convince Republican primary voters that they are the person in the field most committed to addressing long term debt and spending issues.

For that group of voters, it’s virtually impossible to call for too many cuts, to remove the government from too many aspects of peoples’ lives.

No one running for the Republican presidential nomination wants to get crosswise with the party base — and, in particular, the tea party — on the issue. One needs only look to the public denunciation of Gingrich by the likes of conservative commentator Rush Limbaugh and South Carolina Gov. Nikki Haley to know why.

“One candidate’s shakedown leaves the party shook up,” said Tucker Eskew, a longtime party strategist, of the Gingrich episode. “The reckoning was coming anyway, but this episode starkly reminds campaigns that entitlements are on the chopping block.”

The calculus of the 61 House Republicans sitting in districts that President Obama won in 2008 is vastly different. They need to worry about wooing independent voters who have proven to be decidedly fickle over the past four years. (Independents favored Democratic House candidates by 18 points in the 2006 midterms and Republicans by 19 points in the 2010 election.)

And, at the moment, Medicare is a losing issue for Republicans among unaligned voters.

In the final Siena Research Institute poll on the New York 26th district special election, roughly one in four independents said Medicare was the most important issue in deciding their vote. Among those independents, Hochul led state Assemblywoman Jane Corwin (R) by eight points.

It’s hard to imagine that the Republican presidential candidates will moderate their position on the Ryan plan anytime soon — if anything, they are likely to double down on the cost cutting provisions contained within it.

The challenge then for congressional Republicans is to find a way to sell the plan to independent voters in a way that, if the New York special is any guide, they just haven’t done yet.

“If Ryan really wants to force the needed change and not make it a huge political liability he’s got to sell the problem,” advised one senior Republican House strategist who was granted anonymity to speak candidly. “That will take more than a week of town hall meetings and a couple of web videos.” Added the source of House Republicans: “At this point, they’ve bitten off more than they can chew.”

The other thing Republicans must hope is that the presidential candidates propose their own debt and spending plans that push the Ryan plan out of the news.

Of course, with a wide-open Republican field, it’s not clear whether a proposal put forward by any of the candidates — up to and including the frontrunning Romney — would drive the Medicare provisions of Ryan’s plan out of the political consciousness of swing voters.

“I think we need to be more aggressive in defining that Obamacare cuts Medicare,” said Christian Ferry, a senior Republican consultant. “Then, if the Ryan plan is our position for going forward, we need to actually get out there and define what it is and what it is not in an aggressive, offensive manner, rather than allowing the Democrats to play gotcha, scare politics as they did in NY-26.”

The coming days and weeks will be critically important for Congressional Republicans to address their Medicare problem.

The Senate is preparing to vote on the Ryan budget on Thursday and there are already signs of cracks in GOP unity as Maine Sens. Olympia Snowe and Susan Collins as well as Massachusetts Sen. Scott Brown have said they will vote “no” on the proposal.

Any large-scale desertion of the legislation will further add to the concerns among vulnerable Republicans about what the Ryan plan might mean for their re-election prospects. Of course, it’s worth noting that President Obama’s budget plan is expected to get far fewer votes in the Senate than the Ryan plan, a fact GOP strategists will likely seize on to blunt the potential fallout from the failure of the House budget plan.

Simply put: Republicans need to find a way to win on Medicare — or at least fight it to a political draw — as soon as possible. Otherwise, they run the risk of it emerging as a major issue in the 2012 campaign.