= Commuting squares of ordinals
{title2=$\alpha\beta=\beta\alpha\iff\alpha^2\beta^2=\beta^2\alpha^2$}
Two <ordinals> commute under <ordinal multiplication> if and only if their squares commute. One implication follows from <associativity>. For the converse, squaring is strictly increasing: $A<B$ implies $A^2\leq BA<B^2$. If $\alpha\beta<\beta\alpha$, monotonicity gives
$$
\alpha^2\beta^2\leq(\alpha\beta)^2<(\beta\alpha)^2\leq\beta^2\alpha^2,
$$
a contradiction to commuting squares. Swap the factors to exclude the reverse inequality; zero factors cause no exception.
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