When I was sworn in to the Army, I had my right hand up, left hand at my side.
When I was sworn in as an attorney, I had my right hand up, left hand at my side.
When I was sworn in as a prosecutor, I had my right hand up, left hand at my side.
When I was sworn in as a magistrate, I had my right hand up, left hand at my side.
It's the oath that matters, not where the left hand is. Assuming that the person taking the oath has any real honor.
That said, if I was standing there being sworn in as President, I'd want that big old Bible that many previous Presidents have used. For the history and the tradition.