  We revisit the fermionic parton approach to S = 1/2 quantum spin liquids with
SU(2) spin rotation symmetry, and the associated projective symmetry group
(PSG) classification. We point out that the existing PSG classification is
incomplete; upon completing it, we find spin liquid states with S=1 and S=0
Majorana fermion excitations coupled to a deconfined Z2 gauge field. The
crucial observation leading us to this result is that, like space group and
time reversal symmetries, spin rotations can act projectively on the fermionic
partons; that is, a spin rotation may be realized by simultaneous SU(2) spin
and gauge rotations. We show that there are only two realizations of spin
rotations acting on fermionic partons: the familiar naive realization where
spin rotation is not accompanied by any gauge transformation, and a single type
of projective realization. We discuss the PSG classification for states with
projective spin rotations. To illustrate these results, we show that there are
four such PSGs on the two-dimensional square lattice. We study the properties
of the corresponding states, finding that one -- with gapless Fermi points --
is a stable phase beyond mean-field theory. In this phase, depending on
parameters, a small Zeeman magnetic field can open a partial gap for the
Majorana fermion excitations. Moreover, there are nearby gapped phases
supporting Z2 vortex excitations obeying non-Abelian statistics. We conclude
with a discussion of various open issues, including the challenging question of
where such S=1 Majorana spin liquids may occur in models and in real systems.
