In Ubuntu, for example, you can use Super-M to invert the screen colors (requires desktop effects, see [1]). Other OSes have similar abilities, although I'm not really concerned with anything other than Linux.
Which API should I look into if I want to write a simple program that, when run, inverts the screen colors?
The language that I use the API from doesn't particularly matter to me. I am familiar enough with C/C++/Bash/Perl/Python that I can hack this up in whatever language has the easiest access to this API. Working on Ubuntu is required, working on other similar *nixes is not terribly important.
[1] https://help.ubuntu.com/community/KeyboardShortcuts#Desktop%20Effects%20enabled%20shortcuts