+1 to everything 72fretter said.
I'm currently running new stock Svetlana 12AX7's & EH EL34's. I've tried a few different tubes in V1 and depending on what's in there it will get a little darker/warmer (I expect that will be the case with JJ's). The amp's heritage is always there though - the clean channel has a modern feel to it. The brilliance will take some of the edge out of it. I think the clean channel sounds great with my SG in the middle position, and will do a spanky bluesy clean in the neck position, especially if you push the volume up a bit (and the volume isn't too much).
I'm running it through my V30 2x12 at the moment. I actually like it better through my Eminence Red Fang cab - more balanced & articulate. Something like a G12H30 or a Greenback might be a better option if you want a little darker tone. I've also tried it with my Weber Thames loaded 2x12 (70's Fane "purpleback" clones) and it's a little too "hi-fi" for my taste, especially on the gain channel.
With all that said, the amp is more relaxed in low power mode. It's a very different feel.
Depending on what rectifier tube you use it could make a difference also. Different tubes will deliver different B+ voltages. I currently have a JJ in mine and the tube recto B+ voltage is close to the SS recto. The tube recto mode is somewhat softer/saggier, but not a whole lot. With a different tube the difference may be more pronounced.