  We show that the confining property of the one-gluon propagator, in Coulomb
gauge, is linked to the unbroken realization of a remnant gauge symmetry which
exists in this gauge. An order parameter for the remnant gauge symmetry is
introduced, and its behavior is investigated in a variety of models via
numerical simulations. We find that the color-Coulomb potential, associated
with the gluon propagator, grows linearly with distance both in the confined
and - surprisingly - in the high-temperature deconfined phase of pure
Yang-Mills theory. We also find a remnant symmetry-breaking transition in SU(2)
gauge-Higgs theory which completely isolates the Higgs from the
(pseudo)confinement region of the phase diagram. This transition exists despite
the absence, pointed out long ago by Fradkin and Shenker, of a genuine
thermodynamic phase transition separating the two regions.
