No. The op-amp inputs can not be exchanged.
Why? Otherwise, the feedback loop that ensures the Bandgap stability will not work.
For example:
If the output voltage increases (for any reason), the current increases, and consequently, there is a increase in the voltages at Vx and Vy. The increase at the negative
Input will be higher, since there is higher impedance in this node. Accordingly, the increase in the negative input will decrease the output voltage. So, you notice that the feedback acts in order to keep the output constant. What is important for the Bandgap operation.
If you exchange the inputs, this feedback I described will not work.
Regards,
Palmeiras