As a result I always try to go beyond expectations and pay a lot of attention to communication.
Some of the things I do which I think give my employers confidence in me are:
1. Constant communication. Don't leave it to the imagination what I'm up to. I have a weekly work diary where I say what I'll work on, then tick off what I have worked on. I update throughout the week via IM or email.
2. Think about problems no one is working on yet. Kind of "skating-to-where-the-puck-will-be". In my case as a product designer I work on features/ideas we haven't discussed yet (usually not on company time).
3. Flexible with time. Need a presentation done over the weekend? Sure, no problem.
I'm a huge fan of remote working. My ideal scenario is 4 weeks home, 1 week office or something along those lines depending on projects. Although currently I have not been to the office in about 1 year!