Double Lattices and Covering

One reads online that: A double lattice is the union of two Bravais lattices related to one another by a point reflection.

Moreover, a point reflection in 2D is the same as a 180 degree rotation.

Double lattices are often considered when looking for closest packs of the plane by a given convex region. For example:

https://mathoverflow.net/questions/285580/thinnest-covering-of-the-plane-by-regular-pentagons

https://mathoverflow.net/questions/256351/terrible-tilers-for-covering-the-plane/256493#256493



To put things precisely: for packing copies of a convex region R in 2D, it is usually useful to have one lattice of translates of R and another lattice of translates of -R. This is a practical observation. Doubts: In 2D, what is so special about 180 degrees? Given a convex region R, if one considers two Bravais lattices of unit Rs related to each other by rotation about some other angle, is it guaranteed to give a poorer pack than some two lattices of R's related to each other by 180 degree rotation (the minus sign in -R)? And further, do double lattices being good candidates for 2D packing automatically make them good candidates for thinnest covering too?

Shall ask around and add to this post...