10. A regular tetrahedron has three side edges of length 1, and they are pairwise perpendicular. If this tetrahedron is rotated $60^{\circ}$ around its height, then the volume of the common part of the rotated tetrahedron and the original tetrahedron is
10. $\frac{1}{9}$.

The common part is a regular hexagonal pyramid with a base area equal to $\frac{2}{3}$ of the original base area and the height unchanged.