Both Major Major and Yossarian would routinely sign official documents and censorship documents with "Washington Irving" (I have another question on exactly why they do that). When that became too monotonous, they'd sign stuff with "Irving Washington" for awhile. Why didn't they simply select a completely different name? Why did they consider reversing the names less monotonous? Or is that part of the joke (the fact that it really wasn't all that different)?
Well, it really isn't all that different, but the bureaucracy is so soul-crushing that that's about the only thing they can do to reduce the monotony.
Yossarian did actually occasionally sign different names - for example, in one case, he signed as the Chaplain.
On the other hand, part of the advantage of using that particular name (for Major Major, at least) was the fact that stuff signed with that particular name never came back to him, so I'd imagine he would probably be reluctant to try entirely new names (given that he found one that "worked") in case he got hassled to re-sign the same papers over and over again as a result. Reversing the name provided at least the illusion of variation and autonomy with much lower risk of getting the documents back again.