I'm pretty sure that happened a lot back in the days. In fact, I believe at one point Walter Johnson started three in a row against the yankees. He often pitched out of the bullpen one night for a couple of innings and pitched a complete game the following day. Or pitched a complete game one day and came out of the bullpen the next day to help out. This lead to some unfair losses credited to his overall record on top of having little offensive help. He pitched 8.1 one day in Chicago for the win and the next day in St Louis he came out of the bullpen for .1 and the loss. He pitched two innings out of the bullpen in Boston and took the loss only to start the second half of the doubleheader and went 9 innings to secure the split in the doubleheader. Later that year (1923) in St Louis he pitched 3 innings out of the bullpen and again started the second game of the twinbill and pitched 7 more innings to sweep the Browns. I read about a game he pitched in NY where he pitched 13.1 innings and lost on a walkoff. He had many extra inning losses and no decisions. But also had lots of extra inning shutouts. 
One year he got pounded by Detroit at home in 4.2 innings but came back the next day and tossed a complete game yielding 3 ER's and taking the loss.
Babe Ruth also started both halves of a doubleheader once but only pitched .1 of an inning in the first game to allow Rube Foster more time to loosen up. He pitched a complete game in the second game. It's possible that he also started back to back games when he was tossed from a start after facing only one batter for hitting the umpire but I can't find any records for that year.
...or Joe McGinnity (active in MLB 1899-1908) who on five different occasions pitched both ends of a doubleheader - and three different times won both matches.
What a different game it was....
