Factory Reset

I don't believe this is possible to my knowledge.  You would simply need to program over the old settings with your new desired settings.  Is there a problem in particular you're trying to solve?

Yes, when I first set it up I programmed the All-On Power feature. It never quite worked correctly. It would turn off the cable box instantly but the TV would turn off AND back on. I tried holding the power button for different intervals. If I got it just right, maybe one chance in ten, it would turn off both, but with a problem I'll detail below.

If I select the TV button and press power just once the TV will turn off and on two or three times, sometimes going off but most times staying on. About three tries of the power button does the trick but I'm concerned the cycling off and on before it shuts down may somehow damage the TV. So I just started turning the box off by itself by selecting the cable button and pressing power then getting up and turning off the TV manually. When I turn off the TV manually it goes right off, no cycling, as it does if I use the TV's own remote.

I want to turn off the All-On Power feature to see if it will fix this problem by powering off the box and TV individually using the cable and TV buttons on the remote.

I see on the Cox support page there is a factory reset procedure for the Cox Checkmate III, RT-U27 remote so I was hoping there would be one for my URC-8820. Or at least find a way to turn off the All-On feature.

The 9xx code for Restore Factory Defaults is 981.