  In this paper, which is a complement of \cite{BG}, we study a few elementary
invariants for configurations of skew lines, as introduced and analyzed first
by Viro and his collaborators. We slightly simplify the exposition of some
known invariants and use them to define a natural partition of the lines in a
skew configuration.
  We also describe an algorithm which constructs a spindle-permutation for a
given switching class, or proves non-existence of such a spindle-permutation.
