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What does " other way around" mean?

Depending on context, it means either the opposite position, direction, order or situation.

It can be concrete:
Is John taller than Jim? No, it's the other way around: Jim is taller than John.

Or abstract:
Lester Siegel: The saying goes, "What starts in farce ends in tragedy."
John Chambers: No, it's the other way around.

It means to reverse it. Salt to the right, pepper to the left. No no, other way 'round. Oh so you mean Pepper to the right, and salt to the left.

So it can mean position, direction, location, the order (this is first, no other way around, this is actually last), or even ideas or thoughts.

Wealthy countries are the happiest, no its the other way around, Wealthy countries are the unhappiest.

It's like turning your sentence around, like you would turn a shirt inside out. It doesn't always literally mean go in the opposite direction, it means to go with the other option (or way), however a way is often a direction, but it can be many other things (ways of being, ways of living, ways of thinking)

I always thought black bears were dangerous, and grizzlies friendly.
No its the other way around, grizzlies are friendly and black bears are dangerous.

It's also prevalent in balanced sentences:
To everything there is a season,
I time to laugh, a time to cry
I time to break down, and a time to build up

It's like this only you are literally using other way around in a comparative, or direction based correction or order. Think of it as correcting the steps in a recipe book. No not that, way, other way around, the flour goes first, then the butter, not the butter then the flour.