Party politics evolve slowly, but it's difficult to get two parties to actually switch places like the Republicans and Democrats did, because most of the time this involves both of them having to pass through a middle ground - the OTL switch of the Reps and Dems on "social issues" was mostly the result of the Dixiecrats placing so much import on [lack of] Civil Rights, and the Republican leadership's decision to take advantage of the situation and court them heavily (the famous "Southern Strategy").
But it's harder to abruptly about-face on economic policy, which is more universal and fundamental to both parties (at least, historically). Without yet touching on history, I would see the process as likely going something like:
1. Republicans start getting more and more economically left, Democrats remain there, American economic discourse and policy just starts migrating leftwards
2. American economic policy is now broadly leftish
3. In response to some internal pressures, the Democrats start shifting liberal economically, until eventually they're roughly where modern Republicans are and the polarity is switched
Or, of course, it could go the other way around.
I think we actually almost saw something like this. The 90s saw Clinton leading the Democratic Party to more and more liberal economics with a general feeling of deregulation and free trade (specifically, for example, NAFTA and the repeal of Glass-Steagall). Starting under the Tea Party, the Republicans began changing from a party controlled by its Rockefeller-type magnate overlords into a much more populist party. While OTL saw the rise of the Tea Party linked pretty closely to Obama, the rise of the populist core of the Republican Party was actually a long time coming and would likely have occurred during tough economic times regardless of the particular president.
So let's say something like, Gore wins in 2000 and continues the Democratic Party's push towards increasing liberalization. Environmental laws stay on the books, but free trade and unregulated markets are the names of the games, and enthusiasm for free trade as China starts taking jobs from even formerly secure manufacturing sectors like technology eventually causes some unions to get angry. Unions start to break with the Democratic Party, and the Republicans are smart enough to figure out a way to court them. As the Democrats continue to push "right" economically, poor white Republicans, led by the new unions and nativists, start agitating to protect American jobs. The housing bubble, which was gonna pop no matter who was president, does so, and lower-middle and middle-middle class white Americans are the biggest victims (as OTL), and a grass-roots Tea-Party like group starts calling to protect good Americans from Wall Street predators (it's more credible than the Tea Party, though, because there's no way to accuse it of being racist or hate-driven).
When the dust settles by 2015 or 2020, we get the seemingly paradoxical positions whereby the Republicans call for less taxation but more regulation and the Democrats more taxation but less regulation.