This is a community wiki answer based on the comments by alesia and Richard Stanley.

You already gave the hardest part of the argument. To finish it off, note that translating by $\mathbf{l}$ the Voronoi facet dual to edge $(\mathbf{0},-\mathbf{l})$ gives the Voronoi facet dual to $(\mathbf{0},\mathbf{l})$ because it is an isometry and a symmetry of the lattice.

That every facet of a Voronoi polytope is centrally symmetric is mentioned in the paragraph after Theorem 1.1 of R. M. Erdahl, Zonotopes, Dicings, and Voronoi’s Conjecture on Parallelohedra.