1080p versus 1080 Input Capability

It seems that most units that can display 1080p don't accept input of 1080p, but take a 1080i input, deinterlace and then upconvert to 1080p. However, Sharp specifically makes the point that the larger panels of their Aquos line accept native in put of 1080p. My Helios X3000 upconverts its signal inputs to 1080p which then feeds into the TV. The Helios also could feed a 1080i signal to the TV.

Does it make a difference in picture quality whether the deinterlacing and upconverting is done by the DVD or the TV?