Suppose I have an oriented, compact manifold-with-corners $M$ (see Jack Lee’s Introduction to Smooth Manifolds, pages 417-419). It is not hard to see (using the results of these pages) that its boundary $\partial M$ can be written as a finite union $\partial M=\bigcup_{i=1}^n H_i$, where each $H_i$ is a manifold-with-corners. $M$ being oriented, each $H_i$ is oriented as well via the induced orientation. That is, each $H_i$ is an oriented, compact manifold-with-corners, and hence its boundary can be decomposed as a finite union of manifolds-with-corners as well. We can thus write $\partial H_i=\bigcup_{j=1}^m D_{i,j}$, where each $D_{i,j}$ is an oriented, compact manifold-with-corners.
We can take the free abelian group freely generated by $\{D_{i,j}\}$, where we identify $D_{i_0,j_0}$ with $-D_{i_1,j_1}$ if the two are the same manifold (i.e., the same subset of $M$) but with reversed orientations. We may then consider the sum $\sum_{i,j} D_{i,j}$, which may be considered as the “boundary of the boundary” of $M$. Is it necessarily zero?
In other words: is it necessarily true that $\{D_{i,j}\}$ can be written as a union of pairs, where each pair consists of the same manifold-with-corners, only with reversed orientation?