OK, I'm a complete idiot. <_< Apparently when transporting the guitar the volume knob on my guitar must have gotten turned to 0 (or something very close). When powering the POD up, it sets the Variax volume based on the preset. When pressing the knob (engaging the Variax models) the "soft" settings get overwritten, and the actual knob positions are read, making everything muted.
Since the way I have things set up I pretty much never play around with the volume knob, it never crossed my mind that this could be the problem. Do I feel silly now... -_-
<does some running head first into a wall>
EDIT: At least in my panic mode, I managed to update the Variax firmware to the latest. :P
OK, I figured it out. It's the L6 Link setting that was mis-configured. Apparently it has a number of options - Amp A/B/A+B and L/R/L+R. I had it set to Amp A. I'm still a bit unsure why would anyone need this setting (I guess it's needed when multiple DT amps are used?) but switching to L+R solves my issues.
Shaamaan's post in Dream Rig stops working when enabling effects on Variax was marked as the answer