Running along the list of unanswered questions, I came along the question about different EDs each episode. And I wondered, has there been an anime that actually switched its OP and ED during the runtime of the show? And if so, which one(s) would that be?
This is not uncommon - a number of shows will have a first episode with no song at the beginning, and with the OP at the end of the episode. I'll pull in some examples of this later.
There are also somewhat more unusual cases like Madoka (as Memor-X points out in the comments), in which "Connect" (usually the opening) was used as the ending song in episode 10.
But, for a particularly amusing example of this, look at Hanamonogatari. In the Monogatari series, OPs are always sung by the voice actors, and EDs are sung by other singers. Hanamonogatari's OP was "the last day of my adolescence" (by Kanbaru Suruga / Sawashiro Miyuki), and the ED was "Hanaato -shirushi-" (by Kawano Marina).
Since Hanamonogatari was broadcast as a continuous ~2-hour long special, the OP and ED each only ran once - but the ED was played at the beginning, and the OP was played at the end! Imagine that!
(I think this is because the OP visuals and lyrics are spoilery for Hanamonogatari, whereas the ED visuals/lyrics aren't? But I'm not sure about that.)
This questions seems answered to me, but since it isn't accepted yet, maybe I can add some more info.
From this question I have a local copy of the database powering ANN's encyclopedia. So with this in hand I looked for titles, that had the same songs as both OP and ED (at some point, not necessarily the same episode).
All title comparisons are done case insensitive to catch annotation inconsistencies. There are no checks for episodes - as long as an OP becomes and ED or vice versa at some point during the anime, it counts (once). The code can be found here.
If we only take the song's title independent of the artists performing it, there are 233 cases from 216 animes. You can see the full list of titles here. If we only consider cases where not only the title was swapped, but it was also performed by the same artist, there are still 210 cases from 194 animes (full list here). In other words, there are 23 titles that not only swapped positions, but at the same time also got a new artist. Then again, upon manual inspection only 14 of these cases turned out to be true, the rest can be attributed to some kind of inconsistent annotation by ANN.
It is probably safe to assume there are more errors, but the numbers should at least be roughly correct. I did my best catch all of the systematic inconsistencies. (The results also confirm the three cases mentioned in senshin's answer)
Most swaps: The only title to have three songs swapped is Major. The openings Saraba Aoki Omokage, Play the Game and Kokoro e were used as endings in 4 episodes in total, even with the same artists. The full lists for the number of swaps per anime is here (including same artist, title only).