Just came across this post... Here's some more anecdotal information to add to the conspiracy theories...
In addition to the 2 aways that I actually went on, I had communication with the residency coordinator at about 6 other programs regarding an away rotation. All of those programs have offered interviews and I have gone 0-for-6. A couple of those places may have been 'out of my reach', regardless. The other 4 are local and/or 'less competitive' programs that have alumni from my school already in their residency.
For anybody that cares about the details: No, I didn't initially take a shotgun approach to AI applications. I originally applied to just 2 places for away rotations, and did so quite early in the process. One of the programs (a big name in ortho that will remain anonymous) jerked me around until mid-to-late April: told me I should be able to rotate, then only rarely responded to follow-up emails; then subsequently claimed that I didn't turn in a portion of the information that they needed, even though I collected everything myself and sent it in one single envelope; then realized that, oh, I did turn everything in, and forwarded the info 'immediately to the program director' for 'a decision in next couple days to a week'; then kept me in suspense for a couple more weeks before I finally sent another email, which was answered a few days later with a very brief 'thanks but no thanks'.
I panicked a little and sent emails to 7 programs in mid-to-late April to inquire about away rotation availability. None of that communication went so far as to accept any spots, but I did submit all of the necessary application information to each program. I eventually accepted a rotation at the first program that was able to offer me a rotation during the necessary dates, and politely informed the others that I would be rotating elsewhere.
Maybe I got blacklisted at the other 6 program for the above, or maybe it's just coincidence. I'm going to guess it's the latter, and that this entire process just causes us to overthink everything.