Throughout the entire movie, Keanu uses his right hand for just about everything, this is especially noticeable when he tries to shoot Al Pacino in the end. BUT...when his nutty wife into the asylum he signs with his left hand.
There's nothing strange about what you observed, as a "lefty" myself, writing is the only thing I do left handed. I shoot a gun right handed, bowl right handed, swing a golf club right handed, throw a ball right handed...