  In this paper we give a detailed classification scheme for three-dimensional
quantum zero curvature representation and tetrahedron equations. This scheme
includes both even and odd parity components, the resulting algebras of
observables are either Bose q-oscillators or Fermi oscillators.
Three-dimensional $R$-matrices intertwining variously oriented tensor products
of Bose and Fermi oscillators and satisfying tetrahedron and super-tetrahedron
equations are derived. The 3d->2d compactification reproduces U_q(gl(n|m))
super-algebras and their representation theory.
