This actually seems like a pretty simple problem, to be honest, where the obvious solution is the best one.
Inverting a binary tree means reversing the order of the sequence of elements that it represents. Thus inverting a binary tree is simply swapping the left and right edge of every node recursively down the tree.
You could do this with a recursive algorithm by modifying the tree in-place or by constructing a new tree. I don’t see any obvious candidate optimizations other than the fact that it’s straightforward to do in place (e.g. using std::swap in C++, or mem::swap in Rust).
However I don’t think this is as good of an interviewing question as other potential questions. See my other comment in the thread for one that I think is better. The reason is that programming tasks like this are pretty uncommon to encounter as a professional software engineer.
Better questions (IMO) are tasks that professional software engineers are likely to encounter while doing their job at the company — or representations of them, while testing knowledge of A&DS as necessary.
Even if a company does particularly a lot of work with binary trees, then that company would be likely to have number of such A&DS already implemented. If you’re working at Google then I can’t imagine that you would need to build a binary tree data structure or an algorithm for inverting them; undoubtedly these implementations already exist within their standard libraries.
Personally, I can’t immediately ever recall encountering a binary tree in business logic while working as a professional engineer. I wouldn’t expect to unless I was building some thing like a collections library or a specialized solution to a large scale problem like implementing a novel data store of some kind.
And if I was doing that kind of work I’d break out references (e.g. Knuth as a starting point, and any relevant recent research), and review open source software for comparison or as a candidate for directly solving the problem (SQLite, MySQL/InnoDB, LevelDB & BDB, Apache Commons libraries, etc.)