This is a community wiki answer based on the comments by <a href="https://mathoverflow.net/questions/481935/i-believe-that-all-facets-of-a-voronoi-cell-of-a-lattice-are-centerally-symmetri#comment1255084_481935">alesia</a> and <a href="https://mathoverflow.net/questions/481935/i-believe-that-all-facets-of-a-voronoi-cell-of-a-lattice-are-centerally-symmetri#comment1255129_481935">Richard Stanley</a>.

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,
<a href="https://doi.org/10.1006/eujc.1999.0294">Zonotopes, Dicings, and Voronoi’s Conjecture on Parallelohedra</a>.