3:46 p.m. update: House Republicans decided to nix their bid to repeal the medical device tax from their proposal to reopen the government and raise the debt ceiling. 

The key to resolving a more than two-week federal government shutdown and looming debt crisis might just be an obscure tax that most people have never heard of.

We're talking about the medical device tax, a component of President Obama's health-care law that has drawn bipartisan criticism. If lawmakers are able to forge a deal that wins passage in both chambers of Congress, there's growing evidence to suggest the tax could be a key piece of the final agreement.

Let's take a closer look at why.

Neither party is in love with the tax. So why haven't they done away with it in the current standoff? Because of a clash between the House GOP trying to win a key concession on their defining issue, and Senate Democrats sticking to their principled insistence on a "clean" bill with no major string attached regarding the health-care law.

House Republicans have continued to raise the tax in the fiscal talks, even as they have given up on other avenues for weakening the health-care law. They included a repeal of the tax in a stopgap spending bill they passed last month, and now, leaders are talking about including it in a new measure to reopen the government and raise the debt ceiling.

One thing Republicans have working in their favor: Democrats aren't big fans of the tax -- which imposes a 2.3 percent levy on sales for medical device makers in order to help pay for the health-care law -- either. The Senate voted 79-20 to repeal the tax earlier this year. While Democrats have repeatedly shot down the notion of tweaking the tax in the debate over the budget and debt ceiling, it's clear there is opposition to it within party ranks.

Why is the tax such a critical component for the House GOP? Their refusal to vote on a "clean" bill to fund the government and raise the debt limit can be summed up in one word: Obamacare. Cast-iron House conservatives have demanded a fight to shred the health-care law in the fiscal talks, even as it's long been clear that defunding the law or even delaying its cornerstones will never pass the Democratic-controlled Senate.

If a final deal includes a repeal, delay or tweak of the tax, it would serve as the the House GOP's pound of flesh on their signature issue on the standoff. Efforts to make broader changes to the law have gone nowhere and will continue to go nowhere. And smaller changes have been written off by conservatives as merely token gestures.

For Democratic leaders, it's the circumstances of the current debate more than anything about the tax itself. If they bend on the issue, it would represent a concession in the budget and debt talks that they have repeatedly said they won't make. Democrats have long vowed to hold firm in their position that Republicans must give up their efforts to make changes to the health-care law, with, as Senate Majority Leader Harry M. Reid (D-Nev.) has put it, a "gun to our head."

"Is Senator Reid so blinded by partisanship that he is willing to risk default on our debt to protect a ‘pacemaker tax’ that 34 Senate Democrats are on the record opposing, and he himself called ‘stupid’?" asked Boehner spokesman Michael Steel.

Given their hard stance, it's unlikely that Senate Democrats will agree to a repeal. Something smaller like a delay or other tweak seems more likely, though even that could be tough for Democrats to swallow. And on the other end of the spectrum, there are some conservatives who will grumble that even tweaking the tax isn't enough.

But if it means that a broad emerging bipartisan Senate deal can win passage in the House before the nation reaches the point at which default becomes a serious risk, it just might be enough to persuade leaders of both parties that it is the best -- and maybe only -- way out of a deep, deep hole.

WATCH: What is the medical device tax? (Sept. 30)