Good to hear this isn't typical. I didn't think it was!
As mentioned, the set up is simple. I'm just connecting the upconverter to the downconverter with a small shielded SMA cable. I'm generating a single tone at 10MHz using the GenerateSingleTone.vi or something similar to that (see IQPhase.jpg attached). In IQPhase.jpg there is a simple program I made similar to one of the "Getting Started with I and Q" example VI's. Plotted is I, Q, Magnitude, and Phase (in degrees, calculated from I and Q). I took 100ms of data...and you can clearly see the modulation.
The second attachment (spectrum.jpg) is a screen shot of the 5660 Demo Panel. Centered at 10Mhz...a Span of 1kHz...and RBW of a few Hz. The Delta Peak function shows the side bands 60Hz away.
The chassis is being used on a large optical table, which has been sometimes known to act as a big capacitor, picking up all sorts of EMI. However, repositioning the chassis (isolating it from the table, picking it up in the air, etc.) did not solve the problem. It seems like maybe it's getting inside the chassis some other way...though as you mention above, it seems like awfully strange behavior.
I appreciate your insight!