  We complete the classification of symmetry constraints on gapped quadratic
fermion hamiltonians proposed by Kitaev. The symmetry group is supposed compact
and can include arbitrary unitary or antiunitary operators in the Fock space
that conserve the algebra of quadratic observables. We analyze the multiplicity
spaces of {\em real} irreducible representations of unitary symmetries in the
Nambu space. The joint action of intertwining operators and antiunitary
symmetries provides these spaces with the structure of Clifford module: we
prove a one-to-one correspondence between the ten Altland-Zirnbauer symmetry
classes of fermion systems and the ten Morita equivalence classes of real and
complex Clifford algebras. The antiunitary operators, which occur in seven
classes, are projectively represented in the Nambu space by unitary "chiral
symmetries". The space of gapped symmetric hamiltonians is homotopically
equivalent to the product of classifying spaces indexed by the dual object of
the group of unitary symmetries.
