  We continue with the ideas of Bonatti-Langevin-Jeandenans towards a
constructive and algorithmic classification of pseudo-Anosov homeomorphisms
(possibly with spines), up to topological conjugacy. We begin by indicating how
to assign to every pseudo-Anosov homeomorphism an abstract geometric type
through a Markov partition whose rectangles have been endowed with a vertical
direction; these are known as geometric Markov partitions. Such assignment is
not unique, as it depends on the specific geometric Markov partition, and not
every abstract geometric type is realized by a pseudo-Anosov homeomorphism.
This poses interesting problems related to the algorithmic computability of the
classification. The article contains three main results: $i)$ the geometric
type is a total invariant of conjugacy, $(ii)$ a pictorial and combinatorial
criterion for determining when a given geometric type is realizable by a
pseudo-Anosov homeomorphism is provided, and $(iii)$ we describe an algorithmic
procedure for determining when two geometric types correspond to topologically
conjugate pseudo-Anosov homeomorphisms.
