Hauppauge PVR-350 hardware decoder
 

This is probably a really dumb question -

Is the hardware decoder on the PVR-350 routed directly to the TV-OUT, or can you take the output of the decoder, and, say, display through X?

Once you've all stopped giggling, let me know. :)

If all the drivers (ivtv) are setup, then:
/dev/video16 is the device to write the TV-OUT to, and this can be any mpg2 stream or so I understand)
/dev/video0 is the video input... There are packaged scripts to change channels and inputs.