I'm not an engineer, just a nerd with too much time on my hands who's curious... is it possible that your 1230 situation was a chain of two difference products? (1450 - 1340 = 110; 1340 - 110 = 1230?) Or is this "too many steps" from a physics point of view? Since I'm not an engineer, it's FAR more likely that you're right with your 2680 theory, I'm just curious how you went about excluding the other possibilities.
Also, a weird FM/AM story of my own... This may just be an issue with the radio, but if it's not, I think I may have received an honest-to-God SEVENTY-EIGHTH harmonic

off an AM station some years ago. I used to live right in the main lobe of a nine-tower directional AM on 1130. This station was 50 KW daytime (might have only been *pattern* directional on the 25 KW nighttime power, but at the very least the conductivity was better in my direction on the daytime signal.) And I lived all of three miles away, with the towers just barely at eye level on the horizon due to slight terrain differences. Even before it went HD, this station nuked everything from 1060 to 1200 on a Walkman if I stood out on the driveway, with nothing between me and the towers... and sometimes bled into our phone lines.
One early afternoon, I was trying to tune in a college station 80 miles away on 88.1 (theoretically 31 dBu at my location), on a radio which would ordinarily get a weak but listenable signal from it but had recently suffered a broken whip antenna. Instead of Modest Mouse and Guided by Voices, I heard sports talk... To confirm my suspicions, I alternated between 88.1 and 1130 for some time, and sure enough, it was an exact match. There was nothing else on 88.1 in the local area that would have carried the sports format heard on 1130, and a quick check of the car receiver confirmed that there was no tropo or skip in at the time. Like a moron, I didn't listen to the 88.1 interference long enough to get 1130's ID... I figured that the only possibility was their 78th harmonic on 88.14!
In retrospect, I have some uncertainties -- first off, the sound was pretty clear (not strangely garbled, as it intuitively seems would happen if you tried to demodulate something with a different scheme than it had been modulated in), and it even seemed to have some treble well above 5 KHz. Secondly, I did own a rather drifty Ramsey FM-4 kit at the time, but I can't remember if it was on, and it would have been about 100 feet away inside the house (I was outdoors.) So my questions are... is there something about an FM detector that allows it to render "clean" audio from an AM signal it's somehow picking up? Could the events described have possibly originated from a mixing product with the Ramsey kit, if it were on and had somehow drifted to 87.0 or 89.2... even though the thing was only 250 mw TPO and hooked to a (probably poorly resonant) random wire?