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If your position was deemed by the IRS to be that of an employee, the contractor would be the one with a steep tax liability, not you
You will pay your social security/medicare taxes via your schedule C
So your taxes will be paid. They will have to pay their payroll taxes if you are deemed to be an employee
Typically, though, when you have an agreement like the one you signed, you are a contractor and not an employee
Is there anything else that I can assist you with today?