Now that we've taken the octahedron apart again and put each triangle together with the one it was facing in the assembled figure, we have the following pairs:
"100" / "721"
"520" / "310"
"441" / "611"
"301" / "603"
Put each pair together "back-to-back", with the unetched sides hidden. If we line up the outer edges of the pieces precisely, we find that, in one of the three relative rotations, some of the swiss-cheese-like holes will line up nicely.