It took me a while to figure out the mechanics behind how exactly the mood wheel functions so I'll type it down in case someone finds this useful. Or corrects me if I got something wrong :D
The mood wheel shows up at the end of every social turn and will always offer two mood-altering activities. The key part is that it selects those activities by looking at your current mood and always offers to switch to one of your mood's immediate neighbors. So let's say you have the blob pet and visit it during the social turn, which sets your mood to Negative, then you end the social turn. The mood wheel will show that your current mood is Negative and will offer to switch to either Cautious (on its left) or Secretive (on its right). Let's say you pick Secretive and play on, as long as your mood doesn't change during the narrative scenes and the social turn you can choose between Negative and Determined next turn. It's theoreticaly possible to travel the whole wheel like that.
Keeping this in mind helps with planning out bonuses for classes, or for entire playthroughs - for example, if you want to focus on maxing out Stealth this playthrough it's best to pick the blob pet, since Stealth is boosted by Cautious and Secretive and you can force the wheel to offer those every time by visiting blob before the social turn ends. In practice it's a bit more complicated since some narrative scenes can alter your mood before classes so it's not a surefire method, and sometimes you don't have the turn to spare to visit your pet, but it's still really handy.