Remark 7.6.2.8 (Symmetry). Let $\operatorname{\mathcal{C}}$ be an $\infty $-category, let $\sigma : \Delta ^1 \times \Delta ^1 \rightarrow \operatorname{\mathcal{C}}$ be a commutative square in $\operatorname{\mathcal{C}}$, and let $\sigma ': \Delta ^1 \times \Delta ^1 \rightarrow \operatorname{\mathcal{C}}$ denote the commutative square which is obtained from $\sigma $ by precomposing with the automorphism of $\Delta ^1 \times \Delta ^1$ given by permuting the factors. Then $\sigma $ is a pullback square if and only if $\sigma '$ is a pullback square, and $\sigma $ is a pushout square if and only if $\sigma '$ is a pushout square.
More generally, if $U: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ is a functor of $\infty $-categories, then $\sigma $ is a $U$-pullback square if and only if $\sigma '$ is a $U$-pullback square, and $\sigma $ is a $U$-pushout square if and only if $\sigma '$ is a $U$-pushout square.