As stated, it kinda depends on what does it better. The display will take the incoming signal and display at its native rate. Ive always used native and original format, leaves it to one conversion, which is the display accepting 480i/720p/1080i and then displaying 1080p, as opposed to the Directv box doing one conversion, then another within the set.
I now have one of the new Oppo players, which can be used like a video processor, where you run a set top box into it and it then uses the on board Marvell chip to process the signal, like it would a blu-ray. So I have Directv still set to Native, and then the Oppo sending out 1080p/60 to my 1080p display.
My faith is, the top notch video processing of the Oppo will handle the conversion better than the display or broadcom chip in the HDDVR, but if anything it will do just as good a job.
I also have an AVR with an anchor bay chip, and in the past I found (to my eyes) the image looked better when when I used it to convert the Native Directv feed to 1080p for my display, instead of the display doing it. I had handshake problems, but the Oppo is doing it fine with no handshake issues.