  Motivated by the study of ribbon knots we explore symmetric unions, a
beautiful construction introduced by Kinoshita and Terasaka 50 years ago. It is
easy to see that every symmetric union represents a ribbon knot, but the
converse is still an open problem. Besides existence it is natural to consider
the question of uniqueness. In order to attack this question we extend the
usual Reidemeister moves to a family of moves respecting the symmetry, and
consider the symmetric equivalence thus generated. This notion being in place,
we discuss several situations in which a knot can have essentially distinct
symmetric union representations. We exhibit an infinite family of ribbon
two-bridge knots each of which allows two different symmetric union
representations.
