No one could figure out what inverting a binary tree actually means. Like maybe swapping left and right or something?

In which case the best way to invert a binary tree of course being:

struct Node {
 private:
  Node *leftish = nullptr, *rightish = nullptr;
 public:
  static bool inverted;  // I hope no one ever needs two trees...

  // O(1) tree invert operation!
  void Invert() { inverted = !inverted; }
  Node &left() { return inverted ? *rightish : *leftish; }
  Node &right() { return inverted ? *leftish : *rightish}
};
bool Node::inverted = false;

Don't change the tree. Change your perception of the tree.