Question: What is a terrorist?
Answer: Someone who is willing to harm or threaten an innocent bystander in order to achieve an objective.
We can think of the innocent person as being a hostage. We also know that terrorists are willing to harm themselves in an attack.
A central approach to negotiating with terrorists involves getting them to release their hostages. When there are no hostages left, the terrorists have lost most of their leverage.
The Republicans have been holding three main hostages: the middle class, the poor, and the entire US economy. Several times they have been willing to maim or kill the hostages in order to get what they want: destroying the welfare state, protecting the rich, protecting military contractors, whatever.
Here is one frame to understand what is happening now. Obama has simply been trying to release two of the hostages: the middle class and the poor. By protecting unemployment, middle class tax relief, education tax credits, etc., he has stopped the Republicans from threatening to injure these hostages in the next round of negotiations. He has also taken away some the explosives that the Republicans had. Now they cannot use these weapons to force the economy into recession through across the board tax increases. As we know, the wealthy could weather this recession quite well. For the rest of us, it is major threat.
In this sense, the President has gained critical leverage: The only hostage left now is the entire US economy itself (through refusal to raise the debt ceiling) and it involves a Republican suicide attack. They have to take down the whole ship; they can't just threaten to throw the second class citizens overboard while they sit comfortably on the first deck.
In this sense, I believe Obama has gained critical ground in this round of negotiations. Of course we know that the sequester remains to be negotiated as part of debt ceiling negotiations. But remember, military expenditures are tied up in the process. It is as if the Democrats had chained the loved ones of the Republicans (the military industrial complex) with the very hostages that the Republicans want to take down (social spending). Their fates are now inextricably linked. And the only thing that the Republicans have left as leverage is to sink the entire ship. Yes, they are in their first class cabins. But guess what? Seen the Titanic lately? When the whole boat goes down, the wealthy drown along with underlings.