  Tetravalent graphs admitting a half-arc-transitive subgroup of automorphisms,
that is a subgroup acting transitively on its vertices and its edges but not on
its arcs, are investigated. One of the most fruitful approaches for the study
of structural properties of such graphs is the well known paradigm of
alternating cycles and their intersections which was introduced by
Maru\v{s}i\v{c} 20 years ago.
  In this paper a new parameter for such graphs, giving a further insight into
their structure, is introduced. Various properties of this parameter are given
and the parameter is completely determined for the tightly attached examples in
which any two non-disjoint alternating cycles meet in half of their vertices.
Moreover, the obtained results are used to establish a link between two
frameworks for a possible classification of all tetravalent graphs admitting a
half-arc-transitive subgroup of automorphisms, the one proposed by
Maru\v{s}i\v{c} and Praeger in 1999, and the much more recent one proposed by
Al-bar, Al-kenai, Muthana, Praeger and Spiga which is based on the normal
quotients method.
  New results on the graph of alternating cycles of a tetravalent graph
admitting a half-arc-transitive subgroup of automorphisms are obtained. A
considerable step towards the complete answer to the question of whether the
attachment number necessarily divides the radius in tetravalent
half-arc-transitive graphs is made.
