I bought this cable http://www.amazon.com/Micro-Adapter-Cable-Supports-phones/dp/tech-data/B0063J9F6C
It says specifically that it supports the Galaxy Nexus.
Does nothing most of the time I connect it to my phone and TV. Occasionally it will make the home screen display horizontally but still doesn't show anything on the TV. What MHL cables did you get and do they work?