As I was taught, the word order in idiomatic pairs is fixed and should be remembered once and for all. So, the position of the words in an idiom pair can't be reversed, and you cannot say (and a native speaker would hardly ever do it) for example, "reason and rhyme", "tired and sick" or "baggage and bag".
At the same time, revising idiomatic pairs I came across the "gloom and doom" and "doom and gloom" pairs each of which seems to be valid. This made me think that there may be more (I think there are few if any) such idiomatic pairs a couple of which a literate native speaker might suggest off the top of their head.
May it be that these two pairs are different in their usage (not in general meaning) and they are just two separate idiom pairs, each one being used in a different context? I would think so if the example of the usage were not the same as for the one and so for the other:
So what's the ins and outs? Or there really are a few more idiom pairs with non-fixed word order? What are they, if someone can recollect?