I always mix these up when answering since they both have ‘on the contrary’ as a tip when answering.
Is there a good way to tell which one can go in which sentences?
Are they interchangeable or is my understanding of both of them meaning something is contrary to expectations wrong?
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Hey! 
Those two can be used interchangeably when they mean “contrary to expectations”.
逆に cannot be replaced with かえって if you want to say/write conversely, vice versa, backward and that something is reverse in general, without contray to expectations nuance. (in other words, it is broader expression)
女の子はアルファベットを逆に言ました。
The woman was saying the alphabet backwards. only 逆に is ok
母を手伝いたかったが、（逆に・かえって）ずいぶん迷惑をかけた。
I wanted to help mum, but, on the contrary, I caused her trouble. both OK
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