If government institutions are legally bound by the Constitution, then what legal right do public schools have to punish students for hateful comment?

Anonymous
Shouldn't I be able to criticize/curse out the teacher, or make "hateful" comments without consequences? I understand there are some federal laws about hate speech. But if I'm not directing it to a specific individual, just speaking about certain groups of people.

I can give you an example of an incident at MY high school. The LGBT lovers were allowed to have a day of silent protests for LGBT rights, but when a group of rednecks started a KKK faction, they were suspended for 3 days.
If government institutions are legally bound by the Constitution, then what legal right do public schools have to punish students for hateful comment?
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