I don't get why this one is the meme. Just because it's recursion? Because it's (nearly) pointless? There are so many other algorithms I find more difficult/more tedious.

I think it’s a meme because the home brew author tweeted that he was rejected by Google for failing to invert a binary tree.

Is there a practical reason to do this in a real-world program?

Sure, it’s the binary tree equivalent of reversing an array.

For large enough trees it's probably more efficient to instead just switch from preorder to postorder traversal instead of changing the whole tree.