Yeah, the idea that Democrats are center-left is hilarious - by the standards in most of Europe, they're not even center-right, just plain rightwing, whilst the Republicans are pretty much far-right (given their heavy religious, ultra-nationalis, anti-immigrant and warmongering - amongst others - rethoric).
The Overtoon Window has moved to the Right everywhere but in the US it did way much further than in most of Europe.
As for the whole neoliberalism stuff, it's pretty easy to spot the neoliberal parties even when they've disguised themselves as leftwing or (genuine) conservatives: they're the ones always obcessing about what's good for businesses whilst never distinguishing between businesses which are good for people and society and those which aren't: in other words, they don't see businesses (and hence what's "good for businesses") as a means to the end of being "good for people" (i.e. "good for businesses which are good for people hence good for people") but as an end in itself quite independently of what that does for people.