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in the opamp bandgap, there are both positive and negative feedback.
you can not change the polarity of the opamp. because at the startup, the negative terminal voltage is bigger than the positive terminal, so the opamp output voltage will be low, than the circuit will startup.
You should ensure the stability of the bandgap circuit, in this circuit, there are two feedbacks, one positive (negative input of A1 and M1) and one negative (positive input of A1 and M2), and at this condition, the negative feedback extent is deeper than the positive one, which ensures the stability.