Great thanks to Thande of AlternateHistory.com and Kenneth C. Martis for his Historical Altas of the Untied States Congress. Congressional Districts are darn fiddly to do.
It is the middle of the Democratic National Convention, approaching the 8th ballot. A slight change in attitude occurs between the representatives of House Speaker Champ Clark (D-Missouri) and of Tammany Hall. In our timeline Tammany Hall pledged its support to Clarke in the ninth ballot, firmly placing the machine behind Clarke, and forcing our favorite Democratic populist William Jennings Bryan to go with Woodrow Wilson Governor of New Jersey. Tammany Hall's decision would have far reaching consequences, bringing progressivism to the Democratic Party, radically altering the landscape of Europe after WWI, and many others. However, one can imagine that if no endorsement from Tammany Hall had been forthcoming (from some perceived slight by Clark's men, perhaps, or merely a message lost in the shuffle of the Convention Floor), then perhaps Clark's growing momentum would not have been stopped. Wilson was preparing a concession speech to be read out, and so in this alternate world, Wilson concedes after the 12th ballot. Clark is the Democratic nominee, and the Bourbon Democrats and the Urban Machines retain control.
Now, it is likely that Clark would still have won in 1912, and a Conservative Democrat would become president again. But that isn't nearly as fun as another possibility. For, as you all likely know, Theodore Roosevelt had returned to seek the presidency again, and after having been defeated in the convention hall of the Republican in Chicago, split forming a third party the Progressive or "Bull Moose" Party. Now, considering the deep split this engendered in the Republican Party, and a number of other factors, Clark probably would have won. But, but, let us say he does not. Roosevelt was a strong campaigner, and let us have an assassination attempt on Roosevelt (not by James Schrank, that is improbably individually since he was advised to kill Roosevelt, but there were many disaffected people and a culture which encouraged such an act), that engenders sympathy but does not kill Roosevelt (...Roosevelt is on stage, speaking to a large crowd gathered in Michigan, when suddenly a man jumps out from the crowd and dashes towards the stage. Withdrawing a small revolver from his jacket he takes aim and fires off a shot, but not before Roosevelt has come up under him and given a strong punch. The shot flies wildly as Roosevelt disarms the man in a quick tussle using his noted skill in boxing and judo...). All this to one end: A Roosevelt win in 1912.
Really there are 3 other in some ways more interesting scenarios: one where Clark wins, one where Schrank kills Roosevelt (will Taft win on sympathy? Fighting Bob Rise up to win for the Progressives?). and another where Roosevelt never tries to seek the Presidency again and the Republicans remain the Progressive/Left party. But none of them are this fun.
It is 22 years later. Roosevelt 3rd term was... interesting. A semi-disastrous intervention in Mexico, an economic slump and vigorous union competition, Roosevelt winning yet another Peace Prize for his efforts to prevent General European War (Roosevelt ends up as the kind of President better remembered internationally then internally). Defeated in 1916 (what, do you think he'd retire a second time?), Wilson returns triumphant and presides from 1916-1920. However, the Progressives became a strong 3rd party, actually overtaking the Republicans by 1920. The Progressives emerge reenergized by fierce opposition to entrance into a Great War in Europe that breaks out in 1921, which Wilson is drawn into at the end of his second term. Progressives generally feel that if Europe wants to forget what Roosevelt told them the first time, they're welcome to die by themselves (As a consequence of the fact that the Republicans and Progressives win the house and senate in 1922, there's nothing like the Red Scare of 1918 in ATL. The Socialist Party is much stronger, although that isn't visible in the map). Fightin' Bob is elected in 1924 to another... interesting term. The Progressives are the largest party in the Congress individually but the so called "Wolf" coalition of the Democrats and Republicans (Jackson would be proud) stymies him, which conflicts greatly with his style, along with a post-war slump. Democrats return in 1932-1936.
Now US politics settles into a 2.5 oparty system. The Republicans are now mostly a regional party of New England with strong local support in Ohio, Indiana, Utah, and Wyoming. Dedicated to free-trade, balanced budgets, and isolationism, the Republican party could be placed on the "right," but the coalition is still fairly broad if not nearly as broad as it was pre-1912. The Democratic Party is primarily the party of the South, with support from Urban Machines (mostly Catholic) and some western farmers, and they compete for union votes with the Progressives. Democrats are usually in the "center," but again, broad. The Progressives are a diverse coalition of labor, farmers, the bourgeois, clerical workers, radical liberals, socialists willing to compromise, blacks, whites, urbanites and many others. Generally the Progressives are on the left, and push for government economic intervention, personal liberties (they are the only party that is explicitly opposed to lynchings, although this tends to get muddled depending on where the Progressive is from). Progressives get strong support in the West, the old Northwest, and some parts of the East Coast (Governor for life Gifford Pinchot of Pennsylvania is the last big man of the Progressive Party truly connected to Roosevelt) and often acts the second party of the South (see North Alabama, and note that had it not been Gerrymandered, they would have picked up two seats in western North Carolina).
With an economic slump in 1932, the Progressives pick up the Presidency in '34 under Burton K. Wheeler of Montana, and regain a solid lead in the House and Senate. In '34 the gains in the Congress are much more dramatic (the map would be mostly Green from Washington east to Maine, and from Wisconsin to Louisiana), so I didn't show it. '32 is a pretty solid year to show the 1930-1970 consensus of American politics, with Progressives working with Democrats and Republicans for different things. It gets complicated, I assure you.