It was Fearful Symmetry which first explicitly confirmed that the mirror universe was a parallel quantum reality in the vein of "Parallels". At a staff meeting to discuss the appearance of mirror Illiana Ghemor in the prime universe, Julian Bashir explains about Worf's experience in 2370 to everyone else who was unaware and also says that the computer checked mirror Illiana's quantum resonance signature and came up with only one match: mirror Bareil Antos who had crossed over in 2374.(as well as the notion that our Odo may have come from the Mirror Universe).
Really? That's odd. So if our Odo is from the MU, then is the mirror Odo from the regular Trekverse? And how'd they manage to switch places?
I guess it would fit, though, since MU Odo was such an arrogant jackass - like most Founders of the RU were.
I certainly didn't read it as suggesting that such a switch had actually taken place, only that it was as if it had, given the nature of the two Odos and how each was more suited to the other universe's Changeling culture. An observation, nothing more. That said, I guess you could take it as a literal hypothesis, since both Odos came through the wormhole on their way in from Gamma Quadrant, and as this book stresses several times the wormhole is the surest "natural" way to cross between the realities. So, maybe they did switch. Although I didn't assume the novel was saying they had.
I would think that at some point given all their encounters with doppelgangers from the mirror universe, the DS9 crew would have done quantum scans of themselves, including Odo.
So while an intriguing theory, I don't take that section to mean that prime and mirror Odo had been switched.