We study the neutralino sector of the left-right supersymmetric model. In
addition to the possibilities available in the minimal supersymmetric model,
the neutralino states can be superpartners of the U(1)_{B-L} gauge boson, the
neutral SU(2)_R neutral gauge boson, or of the Higgs triplets. We analyze
neutralino masses and determine the parameter regions for which the lightest
neutralino can be one of the new pure states. We then calculate the relic
density of the dark matter for each of these states and impose the constraints
coming from the rho parameter, the anomalous magnetic moment of the muon, b ->
s gamma, as well as general supersymmetric mass bounds. The lightest neutralino
can be the bino, or the right-wino, or the neutral triplet higgsino, all of
which have different couplings to the standard model particles from the usual
neutralinos. A light bino satisfies all the experimental constraints and would
be the preferred dark matter candidate for light supersymmetric scalar masses,
while the right-wino would be favored by intermediate supersymmetric mass
scales. The neutral triplet Higgs fermion satisfies the experimental bounds
only in a small region of the parameter space, for intermediate to heavy
supersymmetric scalar masses.Comment: 31 pages, 8 figures, one table and references added, to be published
in Phys. Rev.