I think there's a simpler solution, with one loop and no recursion, in O(n) time and O(1) space.

     static Node treeToCycle(Node root) {
      Node pseudoRoot = new Node();
      pseudoRoot.left = null;
      pseudoRoot.right = root;
      Node tail = pseudoRoot;
      Node rest = root;
      while (rest != null) {
        if (rest.left == null) {
          rest.left = tail;
          tail = rest;
          rest = rest.right;
        } else {
          Node temp = rest.left;
          rest.left = temp.right;
          temp.right = rest;
          rest = temp;
          tail.right = temp;
        }
      }
      Node head = pseudoRoot.right;
      head.left = tail;
      tail.right = head;
      return head;
    }