 Orientability of factors of a product manifold

For nonempty smooth manifolds $M,N$, orientability of the product manifold $M\times N$ implies orientability of each factor. Fix $q\in N$ and an ordered basis of $T_{q}N$. Contract a positive top-degree differential form on the product with this fixed basis along $M\times \{q\}$. The result is a nowhere-zero top-degree form on $M$. Interchanging the factors proves the result for $N$.

Conversely, pulling back chosen orientation forms from the two factors and taking their wedge product gives an orientation of $M\times N$. The nonempty assumption is necessary if empty manifolds are allowed: a product with an empty factor is empty regardless of the other factor's orientability.