I used to see this all the time. Most often when folks hadn't shot in a while. It was likely a form issue. When not on your form the first thing to go is throwing your arm "out" or away from the body....left hits for a Righty and right hits for a Lefty.
ALSO:..Don't always believe a center shot guage. A modern bow is a technical machine and as such they have a mind of thier own....just because a guage says its in center doesn't mean it will be the center throughout the shot sequence! Trust your arrow hits and adjust from there.

My Tuning sequence....Eyeball the centershot, Get it close by using an arrow on the string and line up. Next PAPER TUNE ...and don't tell me you don't have a tuner. You have a cardboard box? You got a tuner. Just cut a hole through two opposing sides and tape a piece of paper over it tight.
Once paper tuned do a walkback and see if your arrows are drifting is they are, adjust accordingly and then go back to the paper and see what happens...You will be suprised at the how perfect the holes now are...and if not make note of the tear. This is "tuned" with your set up, and you are now done.

We can throw all the "rules" of tuning around all we want but every different bow/rest/arrow spine/nocking mechanism/release and form will change things. Therefore, not all the rules apply to all the setups!!!