  We provide a systematic treatment of the tenfold way of classifying fermionic
systems that naturally allows for the study of those with arbitrary $N$-body
interactions. We identify four types of symmetries that such systems can
possess, which consist of one ordinary type (usual unitary symmetries), and
three non-ordinary symmetries (such as time reversal, charge conjugation and
sublattice). Focusing on systems that possess no non-trivial ordinary
symmetries, we demonstrate that the non-ordinary symmetries are strongly
constrained. This approach not only leads very naturally to the tenfold
classes, but also obtains the canonical representations of these symmetries in
each of the ten classes. We also provide a group cohomological perspective of
our results in terms of projective representations. We then use the canonical
representations of the symmetries to obtain the structure of Hamiltonians with
arbitrary $N$-body interactions in each of the ten classes. We show that the
space of $N$-body Hamiltonians has an affine subspace (of a vector space)
structure in classes which have either or both charge conjugation and
sublattice symmetries. Our results can help address open questions on the
topological classification of interacting fermionic systems.
