Power derives from your credentials, your ability to argue and your results (frequently in that order).
To Handy's original description of four cultures, I've added a fifth: the Academic culture. Companies with academic cultures look like research labs. Your reputation outside the company (how your peers in the industry view you) could count for more that your reputation inside the company. These companies really are the translation of an academic laboratory to a business environment. You might find them in the R&D organization of a large company, but sometimes whole companies will look like this - especially when a scientist has built a startup. Many biotech companies begin as Academic Cultures.
The biggest challenge of an Academic Culture is that they are frequently oriented toward technologies, not products or markets. This may be appropriate (at least for a while) if there is a high degree of technological risk and scientific breakthroughs (like creating a cure for a disease) need to be made to get a product to market. The serious downside is that the company loses sight of the market in developing the technology. Scientists can become enamored of the technology and end up developing a product that solves a problem that no one has, or more seriously, will pay for. Lots of companies have technologies in search of a market.
If you are in a company with a great, technologically-sophisticated product, but no customers, you're in an Academic Culture.
That said, if there is a market focus, and the scientists are willing to forgo great science in order to build great products, these companies can be a great place to work.
Would this culture be a good fit for you? Take this quick assessment to find out.
