  Supercurrent transport is experimentally studied in a Josephson junction
hosting a double quantum dot (DQD) with tunable symmetries. The QDs are
parallel-coupled to two superconducting contacts and can be tuned between
strong inter-dot hybridization and a ring geometry where hybridization is
suppressed. In both cases, we observe supercurrents when the two interacting
orbitals are either empty or filled with spins, or a combination. However, when
each QD hosts an unpaired spin, the supercurrent depends on the spin ground
state. It is strongly suppressed for the ring geometry with a spin-triplet
ground state at zero external magnetic field. By increasing the QD
hybridization, we find that a supercurrent appears when the ground state
changes to spin-singlet. In general, supercurrents are suppressed in cases of
spin doublet ground state, but an exception occurs at orbital degeneracy when
the system hosts one additional spin, as opposed to three, pointing to a broken
particle-hole symmetry.
