Like a combination of OLED and E Ink.
This will also give you the possibility to turn your TV into a painting (without using power) when you are done watching.
If you think about it, that's what an LCD is doing. The backlight is emitting; the liquid crystals are absorbing. The display is white (i.e. passing through the emission from the backlight) where the liquid crystals aren't active; the display is black-ish (i.e. attempting to block as much of the emission as it can) when the liquid crystals are active.
It's sort of like painting on a lightbox with black paint.
Some ebook devices also have ‘backlight’―usually diffused light from the sides so it doesn't shine directly in the eyes. I think this is the crux of your idea, because currently even if you use lcd/oled with backlight, no-power eink will be severely washed out in comparison. And if eink gets good color and contrast in the future, it will then fulfill the two-mode technology.
The page on the wiki.laptop.org isn't too clear on the technology—presumably it's more complex than just LCD and eink slapped together. The Wikipedia page on the XO laptop seems to have some detail: https://en.wikipedia.org/wiki/OLPC_XO#Display
The E Ink layer would be great for status overlays or permanent elements that rarely change, like the date and battery charge, or a list of recent notifications while the "main/active" display sleeps when the machine is idle.