The "mixed episode" part is usually considered a "specifier", that is, something added on to the primary diagnosis of Bipolar Disorder. Rapid cycling is also a specifier, so for example, if your current state were to be described according to DSM-IV nomenclature (which is what is used in the US), then it would read something like: "Bipolar I, Current episode mixed, with rapid cycling". Do you only have mixed episodes? There are people who are considered "ultraradian" cyclers, and cycle multiple times throughout a day. That's an extreme form of rapid cycling. It's the "...current episode..." portion that changes with the mood states, or it can also be listed as "...most recent episode..." if a person is in remission. "Psychotic features" can also be added as a specifier.

Whatever they want to call it, I am glad that you finally have some answers, so that you may now find solutions. Having rapid cycling and mixed episodes isn't as rare as you might think, and there are lots of people here (myself included) who can share in that understanding with you.