Is Mega Man left or right handed?

Well, after looking at official box art, Megaman uses either hand for his arm cannon. For example, here's some Japanese promo art from Megaman 3 depicting the cannon on his left arm.

While here's some Japanese promo art from Megaman 2 depicting the cannon on his right arm instead.

If you check more of the images and box art, there's no real rhyme or reason as to which hand the arm cannon appears on. So, since he uses either hand just as often in the images and in the games, the only conclusion I can draw from this is that he is indeed ambidextrous.

Well... the original Megaman had his buster on his left, and X on the right.

(I am confused each time he turns left or right)

Well, this happens because when the Megaman sprite turns, the entire sprite image is flipped horizontally, meaning depending on which way you're facing, Megaman's buster will be on different hands. This page can tell a bit more of what I just said, though it isn't exactly an informative site : /

But being a robot/android/Navi, I really think Megaman is programmed to be ambidextrous.

In Megaman Legends (and/or Megaman 64), Megaman's buster is on his left hand. Since the game is three dimensional, there's no 'sprite flipping' like in the original sidescrollers. This would mean that he is right-handed, at least in this game, because that's the only hand he has free when a weapon is equipped.