This is the 5242nd most frequent Russian word.
обратном
"Обратном" means "reverse" or "opposite" (in the prepositional case).
The word 'обратном' here is used in the sense of 'opposite', specifying the case where the conditions are reversed.
В обратном случае, я бы согласился.
In the opposite case, I would have agreed.
The word 'обратном' in this context means 'return' and refers to the way back home.
Он думает об обратном пути домой.
He is thinking about the return journey home.
Here, 'обратном' describes the decision being reversed or contrary to the previous state.