  The reduction of the quasi-Hamiltonian double of ${\mathrm{SU}}(n)$ that has
been shown to underlie Ruijsenaars' compactified trigonometric $n$-body system
is studied in its natural generality. The constraints contain a parameter $y$,
restricted in previous works to $0<y < \pi/n$ because Ruijsenaars' original
compactification relies on an equivalent condition. It is found that allowing
generic $0<y<\pi/2$ results in the appearance of new self-dual compact forms,
of two qualitatively different types depending on the value of $y$. The type
(i) cases are similar to the standard case in that the reduced phase space
comes equipped with globally smooth action and position variables, and turns
out to be symplectomorphic to ${\mathbb{C}P^{n-1}}$ as a Hamiltonian toric
manifold. In the type (ii) cases both the position variables and the action
variables develop singularities on a nowhere dense subset. A full
classification is derived for the parameter $y$ according to the type (i)
versus type (ii) dichotomy. The simplest new type (i) systems, for which $\pi/n
< y < \pi/(n-1)$, are described in some detail as an illustration.
