  We consider Hecke symmetries on a 3-dimensional vector space with the
associated R-symmetric algebra isomorphic to the polynomial algebra
$k[x_1,x_2,x_3]$ twisted by an automorphism. The main result states that any
such a Hecke symmetry is itself a twist of a Hecke symmetry with the associated
R-symmetric algebra isomorphic to $k[x_1,x_2,x_3]$. This allows us to describe
equivalence classes of such Hecke symmetries.
