In Texas the answer is absolutely yes. An employer may change an employees job title without knowledge or consent. The only possible constraints on the employers ability in this regard would come from an employment contract between the employee and the employer.
Yes, unless your new job title is libelous per se. e.g. "That guy who sucks at whatever he does" would be quite harmful to your reputation.
Just thought I'd throw that random exception out there - just in case.
An employee at will can be fired for no cause at all. Without a union or an employment contract you have to recourse.
Discrimination based on gender, age, race, religious beliefs etc.. may give rise to a different answer.
Good luck to you.
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