In the below I would have thought 無茶ぶりした時 would have been 無茶ぶりされた時. Am I missing something?
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Both sentences are okay, and mean almost the same thing:
In the former, the subject of 無茶ぶりされる is simply the listener (あんた). In the latter, there is an implicit subject switching; the subject of 無茶ぶりする is the speaker (i.e., あんたは私が無茶ぶりした時こそ必死になって動く). You have to infer the hidden subject from the context.
In general, this is not a rare phenomenon in Japanese. For example, 添付したファイル and 添付されたファイル usually both mean "attached file".
I think what you're seeing here is somewhat akin to expressions like ぼんやりした or びっくりした. 無茶振りをした時 is just expressing the idea that the individual experiences the time as unreasonable. There is no explicit or implicit notion of someone who is being unreasonable or creating trouble for this individual. Rather, it's just expressing a subjective perspective on what this individual has experienced.