Clint Barton (Hawkeye) is an ambidextrous archer. He is seen shooting right-handed in Thor and both left-handed and right-handed in Avengers.

Joss Whedon: Hawkeye also has extraordinary patience.
Jeremy Renner: Yeah, he's a lone-wolf type.
Joss Whedon: [To Downey] Patience, not so much with you.
Whedon: Tony Stark, I mean.
Robert Downey Jr: [Sitting up, turning to Renner] ARE YOU F---ING DONE?! I thought the story should revolve around me from the beginning to the end!
Joss Whedon: This is going to turn into a therapy session.