K3 surfaces of degree six arising from desmic tetrahedra

We study K3 surfaces of degree 6 containing two sets of 12 skew lines such that each line from a set intersects exactly six lines from the other set. These surfaces arise as hyperplane sections of the cubic line complex associated with the pencil of desmic quartic surfaces introduced by Georges Humbert and recently studied by the second and third author. We discuss alternative birational models of the surfaces, compute the Picard lattice and a group of projective automorphisms, and describe rational curves of low degree on the general surface.