Typically, a logo like this is composed of both the symbol and the name, not just the symbol, though sometimes a well-known symbol can function as a stand-alone logo. A branding guideline will tell you the correct arrangement and order when the elements are used together. Often a stacked version is available for a combination symbol/name logo, too, which sometimes fits a truck door better.
The Chase Bank logo consists of the word CHASE plus the octagon symbol to the right of the name. This would never be reversed. Prior to 2005, however, Chase Bank had the symbol to the left. The Bank of Oklahoma consists of the words with their symbol to the left. Bank of America logo is the words with the symbol to the right. I think there is a stacked version, though, for all three of these banks.
Changing the order of the elements is changing the logo.
It's rare that an original logo design includes an optional mirrored version. Mirroring the US flag, though, is commonly done, depending on the context.
In the end, of course, I do what the client wants. My goal is not to force them to use their branding correctly, it's to get the money out of their pocket into mine. Heck, I'll misspell their name if that's what they want.
Brad in Kansas City