Is my school allowed to look up students' tweets!?

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Yes they are allowed to do that because anything you post on the internet is PUBLIC unless you set it to private. So if you and your classmates are dumb enough to post tweets during school then the principal can read those and use them against you! Stop using your phone during school if you dont want to get into any trouble.
1) anything you post online is public and if done during school hours you CAN get in trouble for it and the principal has every legal right to use those tweets against you or your classmates.
2) The principal can impose any punishment he/she wants that is following the school rules. It isn't for us to judge if a suspension is "fair" or not. If that is part of the rules then he/she can suspend anyone for breaking school rules. Don't text at school if you don't want a suspension.