A confusing new argument I'm seeing from the right is that European austerity programs aren't really austerity because some of the deficit reduction has come from tax increases. All that shows is that European politicians favor fiscal austerity, which is a mix of tax hikes and spending cuts—i.e., exactly what European countries have been implementing.
That said, this is a valuable reminder that the debate over austerity in Europe has been mapped onto the American partisan debate in a weird way.
This should all serve as a reminder that in 2009, the American conservative movement took a strange turn on macroeconomic issues. Barack Obama had a stimulus proposal that they didn't like, presumably because it involved spending money on things they thought weren't worthwhile. But instead of saying that, a lot of conservatives started claiming they didn't believe in the entire underlying theory of fiscal stabilization policy. That then led to this whole weird literature on "expansionary austerity" and a lot of proposed misapplications of the findings of that research. The current trajectory in Europe shows that (a) the expansionary austerity literature was nonsense and (b) American conservatives never believed in it in the first place. As has been the case since 1980, the actual American conservative view of fiscal policy is that taxes must never go up. That's an interesting viewpoint, but it has literally no relationship to the austerity debate.
By the same token, a lot of progressives seem to have this mixed up with regard to state budget policy. States need to balance their budgets. Which means that when revenues fall in a recession, they must implement austerity. People can disagree as to whether this should be done mostly with tax hikes or mostly with spending cuts, but it's all austerity.