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such a net wage agreement is valid under German law. Therefore, your employer is entitled to claim the tax returns.
Thanks for providing the income tax certificate. The tax certificate states the gross income as it is the necessary calculation basis according to the German tax law. But that does not change anything regarding the validity of the explicit agreements in your employment contract. A net wage agreement means in Germany that the employer guarantees a certain net income for the employee. That is also reflected in your job agreement (§4 sec. 2) as the employer is willing to increase your gross salary if tax rates are increasing in order to maintain the guaranteed net income. So the right of the employer to claim back any tax returns is the backside of this guarantee and the risk of any tax increases borne by the employer.
The contractual net wage agreement does not have any direct effects on the respective relations to the tax authorities. So you are still liable for the taxes in relation to the tax authorities, however, your employer has contractually promised to pay them for you. The employer needs the power of attorney in order to hand in the tax return forms for you. The wrong tax class etc. will have been corrected with the tax return I suppose. All the details they asked have tax effects in Germany and go into the calculation of the tax return. Nevertheless, you signed a valid net income agreement, so as long as they pay you the agreed net salary, you have no additional claims arising from such tax issues.
your questions now go beyond the original one and the question value offered. I will be happy to review your payslip and answer your question regarding the power of attorney as premium service. However, please note that I am not an tax expert and cannot answer your crossborder tax issues.
They would still be entitled to and could therefore sue you for the tax returned.