  Besides its usual interpretation as a system of $n$ indistinguishable
particles moving on the circle, the trigonometric Sutherland system can be
viewed alternatively as a system of distinguishable particles on the circle or
on the line, and these 3 physically distinct systems are in duality with
corresponding variants of the rational Ruijsenaars-Schneider system. We explain
that the 3 duality relations, first obtained by Ruijsenaars in 1995, arise
naturally from the Kazhdan-Kostant-Sternberg symplectic reductions of the
cotangent bundles of the group U(n) and its covering groups $U(1) \times SU(n)$
and ${\mathbb R}\times SU(n)$, respectively. This geometric interpretation
enhances our understanding of the duality relations and simplifies Ruijsenaars'
original direct arguments that led to their discovery.
