  We study a multi-terminal Josephson junction based on an interacting quantum
dot coupled to $n$ superconducting BCS leads. Using an Anderson type model of a
local level with an arbitrary onsite Coulomb repulsion, we uncover its
surprising equivalence with an effective two-terminal junction with symmetric
couplings to appropriately phase-biased leads. Regardless of the strength of
the Coulomb interaction, this hidden symmetry enables us to apply
well-established numerical and theoretical tools for exact evaluation of
various physical quantities, and imposes strict relations among them. Focusing
on three-terminal devices, we then demonstrate several phenomena such as the
existence of the finite energy band crossings, superconducting transistor and
diode effects, as well as current phase relation modulation.
