You would neither exchange any A for any B, nor vice versa. Then, let's say you knew there was a situation that would allow you to give up A1 to get B2, and you knew that if you did that, you could give up B2 to get A3. Then the refusal to sort your preferences together makes you lose out on objectively climbing up your preferences which are sorted!

I don't understand this argument. The availability of the option of exchanging B2 for A3 changes things. For example, A1 = "get ten dollars", B2 = "horrible torture", A3 = "get twenty dollars". I'd just do the two exchanges, winning twenty dollars without torture. Does this mean torture is better than ten dollars?

So please provide a better argument why every pair of events must be commensurable.