  Benkart, Sottile, and Stroomer have completely characterized by Knuth and
dual Knuth equivalence a bijective proof of the conjugation symmetry of the
Littlewood-Richardson coefficients. Tableau-switching provides an algorithm to
produce such a bijective proof. Fulton has shown that the White and the
Hanlon-Sundaram maps are versions of that bijection. In this paper one exhibits
explicitly the Yamanouchi word produced by that conjugation symmetry map which
on its turn leads to a new and very natural version of the same map already
considered independently. A consequence of this latter construction is that
using notions of Relative Computational Complexity we are allowed to show that
this conjugation symmetry map is linear time reducible to the Schutzenberger
involution and reciprocally. Thus the Benkart-Sottile-Stroomer conjugation
symmetry map with the two mentioned versions, the three versions of the
commutative symmetry map, and Schutzenberger involution, are linear time
reducible to each other. This answers a question posed by Pak and Vallejo.
