The logo for Netflix's House of Cards shows an upside-down & mirrored American Flag with 0 stars on it.
What does this logo represent in terms of the show?
Is it meant to be symbolism with the themes of the show, or just simple dramatic flair?
A flag being flown upside down is a symbol for distress. As far as the no stars I'm looking into that.
While Kevin Howell is quite spot-on in his explanation of the upside down flag representing distress, I think especially his link to the Flag Code gives another, maybe even better fitting explanation. The country itself is objectively seen not in distress during the course of the show. So while the upside down flag could represent some kind of metaphorical distress in the way that the government and the political system are depicted, what may better fit here is the other, albeit invalid, reason for an upside-down flag, disrespect.
By showing the flag upside-down without an imminent presence of "dire distress in instances of extreme danger to life or property" it shows disrespect to the flag and the political ideals (and ultimately the country and its people) it stands for. Of course, I'm neither saying the show or its makers are disrespecting the U.S. flag nor am I making any personal judgement about such behaviour, just that the characters and the way politics are shown have lost the ideals that flag was once supposed to represent. This disrespect is shown by the political schemes and plots that (not only, but most prominently) Frank Underwood plays with the governmental forces (and ultimately the country) in order to feed his own lust for power. The show depicts a political system and moral that have been (in/per)verted upside-down in the same way as the flag that represents them.
The missing stars could either be just some kind of stylization, but could as well refer to something deeper. One might interpret that into the same disrespect symbolization as the upside-down part, but could maybe also come up with something based on the fact that the stars represent the states. Maybe this way it shows that the show isn't about actual politics with any significance to the whole country, but about a man not interested into anything outside of Washington D.C. and misusing the country's fate only as a device for his own goals.
Umm... it represents the fact that you can't legally use the American flag as part a TV show's logo-lockup. So, the art director had to alter the flag enough so that it doesn't technically look like the US flag.
...And then he/she decided to turn it upside down because they realized reading the flag code that it meant distress, and figured it would be a great conversation piece for fans of the show to read into it.