  We introduce collective geometric phases of bosons and fermions interfering
on a linear unitary multiport, where each phase depends on the internal states
of identical particles (i.e., not affected by the multiport) and corresponds to
a cycle of the symmetric group. We show that quantum interference of $N$
particles in generic pure internal states, i.e., with no pair being orthogonal,
is governed by $(N-1)(N-2)/2$ independent triad phases (each involving only
three particles). The deterministic distinguishability, preventing quantum
interference with two or three particles, allows for the genuine $(N\ge
4)$-particle phase (interference) on a multiport: setting each particle to be
deterministically distinguishable from all others except two by their internal
states allows for a novel (circle-dance) interference of $N\ge 4$ particles
governed by a collective $N$-particle phase, while simultaneously preventing
the $R$-particle interference for $3\le R\le N-1$. The genuine $N$-particle
interference manifests the $N$th order quantum correlations between identical
particles at a multiport output, it does not appear in the marginal probability
for a subset of the particles, e.g., it cannot be detected if at least one of
the particles is lost. This means that the collective phases are not detectable
by the usual "quantumness" criteria based on the second-order quantum
correlations. The results can be useful for quantum computation, quantum
information, and other quantum technologies with single photons. \end{abstract}
