We derive general results for the mass shift of bound states with angular momentum ℓ≥1 in a finite periodic volume. Our results have direct applications to lattice simulations of hadronic molecules as well as atomic nuclei. While the binding of S-wave bound states increases at finite volume, we show that the binding of P-wave bound states decreases. The mass shift for D-wave bound states as well as higher partial waves depends on the representation of the cubic rotation group. Nevertheless, the multiplet-averaged mass shift for any angular momentum ℓ can be expressed in a simple form, and the sign of the shift alternates for even and odd ℓ. We verify our analytical results with explicit numerical calculations. We also show numerically that similar volume corrections appear in three-body bound states.