Rig a 10W DC solar panel to feed back into house power?
As the commentators have suggested, you probably could, but it would not be worth it. The big issue is that the solar panel generates DC electricity whereas your home outlets (and the power grid at large) are AC electricity. 10 watts is not a lot of power. Most home circuits provide 15 or 20 amps at around 120 volts, that equates to 1800 to 2400 watts. Keep in mind that in converting the 8V at 1.25A output of the solar panel into 120v AC you're going to lose a significant amount of that energy. 

Can it be done? Yes. 

How would you do it? Generally speaking you wouldn't. If you HAD to, you might consider something like a Tesla Powerwall scenario where you charge a large battery (over a long period of time) and then use some complicated circuitry to discharge AC power. As others have said, this would take an extremely long time to break even cost wise.