The answer to whether a public school principal conduct a search of your son's phone and pockets any time he sees fit is technically no, although my colleague is correct in noting that the standard to justify this intrusion is low.
The law of the land is that public school students have a reduced expectation of privacy while at school, but the US Supreme Court artfully said that a public school student does not shed his rights when he walks through the school house door.
In New Jersey v. T. L. O., 469 U.S. 325 (1985), the Supreme Court ruled that searches in public schools do not require warrants, as long as the searching officers have reasonable grounds for believing that the search will result in the finding of evidence of illegal activity. This is still good law, but the specific facts of each case may implicate more specific rules for specific situations. Safford Unified School District v. Redding, 557 U.S. 364 (2009) illustrates this and is also good law specific to strip searches at public school; the Court ruled that school officials violated the Fourth Amendment when they strip searched a 13 year old girl based only on a student claiming to have received drugs from that student.
A distinct theme emerges from the above and collecting cases, and all rules applicable to the numerous contexts for governmental intrusion are consistent - the more important and fundamental a citizen's personal or liberty interest to be suffered, the greater the government's interest must be in the conducting the intrusion that courts will require.
Although you do not specifically ask, it does not appear reasonable/worth it for you to pursue recourse by way of a civil action. Even if liability were to follow from these facts, it would be difficult to prove injury to your son. Please remember that it doesn't follow from the principal's incorrect understanding of the standard applicable to the search that the principal lacked an articulable basis that was sufficient to justify the search.
Good luck and have a nice talk with your son about his reduced rights while at school and how same should inform and constrain his decision making. A false sense of security or overestimate of his privacy interests in his cell phone while at school will likely result in poor decisions, and regrettable and unintended consequences.
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So long as the principal has a rational reason - a very low standard -- yes, he can do this. The best practice re cell phones is to leave them at home if you don't want the school to look at the contents.
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