I have a few of those myself. I also have a few UHF converters. I have not experimented with them yet. I assume you may be asking something like this:
The cable box is a frequency conversion device like a VCR tuner, so I wonder if I could tune an ASTC (digital) signal on the "box" and down convert it to channel 3/4 and have my HDTV or STB display on channel 3/4 what I tuned with the box.
One thing Im sure of is this: If the box in question demodulates and then re-modulates on the output channel, its useless for ASTC.