While it's theoretically possible to be type rated on both Boeing and Airbus and fly both 'interchangeably', it would probably be a really bad idea.
There are some pretty fundamental differences between the way Boeing and Airbus systems work - which is better is major subject of debate that I won't go into. But they are different, and switching back and forth would be confusing and potentially dangerous.
In contrast, as glendalegoon noted, the 757/767 have a common type rating, and there is sufficient commonality that it only takes a quick 'differences' course to transition between 757/767 and the 777 (or visa-verse).
Many people believe that the Asiana 777 San Francisco crash last year was caused at least in part because the pilot flying was a long time A320 pilot that was transitioning to the 777 and didn't fully understand or appreciate the differences in the way the two aircraft worked.