lcd display with black background under a semi reflective mirror?
You can turn most simple LCDs into white on black by flipping the polarizer sheet.
This link shows the construction of an LCD.

Here's the relevant picture:



If you can disassemble the LCD and get to the polarizing film (marked F in the picture) you can flip it either horizontally or verticallly then reassemble the LCD.  With the polarizeer flipped, black and white will be inverted.

I found this out accidentally many years ago when I disassembled and reassembled an LCD calculator.  I had to do it all over again to make it display black numbers on a white back ground like it originally did.

Not all cheap LCDs can be reassembled this way for the simple reason that you have to destroy the unit to get it apart.  But there are displays that can be taken apart and put back together, and in those cases you can invert the colors.