Boulder Internal Arts an Yi Zong School: Things i tell my students about Push Hands training:

1. Push Hands/ Tue Shou or Rou Shou/ Rolling Hands or San Shou (?)/ Sticking hands is part of any real Chinese internal martial arts system.
2. Push hands is a drill, not fighting or sparring.
3. The Push Hands drill begins with the idea that we have already closed the distance on our opponent, established contact points, and are attempting to control their center. 
4. Sticking hands will develop close in fighting skill. 
5. A lot of schools will treat Push Hands training as an end unto itself, this is a bad habit, it is a training tool designed to develop a specific skill set in a specific fighting range.
6. Within that range there are numerous skills to train, throwing and sweeping, striking, kicking, joint locking etc.