Degree of a factor swap ($\mathrm{deg}((x,y)\mapsto (y,x))=(-1{)}^{mn}$)

The factor swap ${\mathbb{R}}^m\times {\mathbb{R}}^n\to {\mathbb{R}}^n\times {\mathbb{R}}^m$ has degree $(-1{)}^{mn}$. Moving an oriented basis of the first factor past one of the second requires $mn$ transpositions.