As a materials person, one approach would be a material that can change phases with electronic stimuli such that one phase has no electronic band gap (a conductor) and the other phase has a band gap in the visible range (a semi-conductor). I am unaware of and such material, and should such a material exist it would probably be seeing extensive use in Dubai and Las Vegas if it were economical. The biggest challenge in designing such a material for production is ensuring the phase change material is perfectly flat as though polished in the reflective state, and not too rough in the transparent state, e.g. to have a negligible volume change associated with the phase change.
The closest existing materials I am aware of do not have image reflection, but instead become opaque, and there have been several approaches, including suspended particle, electrochromic, etc (see the link). Such a material does what I've explained except instead of changing band-gaps with phase, simply causes changing organization of ionized or ionizable materials. The issue with modifying this to become reflective instead of merely opaque is again related to roughness. The material must reflect nearly all of the incident light, and must do so with a smooth, as-if-polished surface.
The way most LCDs work is by electrically activated, nematic (threadlike) liquid phase crystals which twist into and out plane with the panel, blocking light as required. None of the material properties change as a result of the process, so the LCD crystals would need to be always reflective, which might create an unusual and distracting appearance when the pixels are "off", though it would act nicely as a mirror when on. How to create a reflective nematic liquid-phase crystal is an interesting question in itself. Additionally, to have a polished mirror effect, the crystals would need to line up perfectly, or the image would be distorted in a distracting way. As it stands with displays, if the crystals are a bit unaligned the display still colors the light passing through effectively.