Assume $M$ and $N$ are two oriented smooth manifold with or without boundaries. Then $M\times N$ is an oriented manifold with corners. Inspired by the theory of cobordism or differential forms, the normalized boundary $$ \partial(M\times N)=\partial M\times N \sqcup (-1)^n M\times \partial N$$ could possibly involve an orientation reversing, like $n=\dim M$ or $\dim M-1$.
What should be the right orientation convention?
Moreover, it is very interesting to ask what is $\partial (M_1\times M_2\times \cdots \times M_k)$?