Are there ordinals $\alpha,\beta$ such that $\alpha+\beta \neq \beta+\alpha$ but $\alpha2+\beta2=\beta2+\alpha2$?

Question

I know that if $\alpha+\beta=\beta+\alpha$, then by associativity $\alpha2+\beta2=\beta2+\alpha2$. Was wondering if the reverse holds?

Edit:

It can't be possible. $\alpha+\beta < \beta+\alpha$. $\alpha+\alpha+\beta+\beta \leq \alpha+\beta+\alpha+\beta < \alpha+\beta+\beta+\alpha \leq \beta+\alpha+\beta+\alpha \leq \beta+\beta+\alpha+\alpha$