  We study matrix product unitary operators (MPUs) for fermionic
one-dimensional (1D) chains. In stark contrast with the case of 1D qudit
systems, we show that (i) fermionic MPUs do not necessarily feature a strict
causal cone and (ii) not all fermionic Quantum Cellular Automata (QCA) can be
represented as fermionic MPUs. We then introduce a natural generalization of
the latter, obtained by allowing for an additional operator acting on their
auxiliary space. We characterize a family of such generalized MPUs that are
locality-preserving, and show that, up to appending inert ancillary fermionic
degrees of freedom, any representative of this family is a fermionic QCA and
viceversa. Finally, we prove an index theorem for generalized MPUs, recovering
the recently derived classification of fermionic QCA in one dimension. As a
technical tool for our analysis, we also introduce a graded canonical form for
fermionic matrix product states, proving its uniqueness up to similarity
transformations.
