Yes, you can split the signal in the beginning and have independent effects chains for each side, and separate mono or even stereo outs for each one. There are a few of us Chapman Stick players that use the HX Stomp like this since the instrument has separate pickups for each set of strings.
Here's the tip on page 29 of the HX Stomp manual:
Okay, this one's for the power users: If you move a Split > Y block all the way left, set its BalnceA to "L100" and its BalnceB to "R100," you can process the L/MONO and RIGHT inputs independently. In addition, if you route the Mixer block to Path B (Send L/R), you can process two instruments simultaneously, each with its own mono input, stereo processing, and stereo outputs!