I'm sorry honey. I do agree with you on this one. I do think your tutoring is more important than anyones' hair. That could be scheduled after she picked you up, or the next day. It shouldn't be that big of a problem.
Why is your mother making it such a big issue? Would she get fired if she rescheduled it? I really doubt it.
I don't know what you tell you honey. It seems that you don't have a very good relationship with your Mom.
Are you in therapy? Can you talk to your school counselor about it? Or if you're not in school, can you talk to your tutor the next time you see him/her? You really need someone to talk to.
I wish you the very best. Please keep us informed, okay? We DO care! Hugs, Lee