  We study the interplay of duality and stacking of bosonic and fermionic
symmetry-protected topological phases in one spatial dimension. In general the
classifications of bosonic and fermionic phases have different group structures
under the operation of stacking, but we argue that they are often isomorphic
and give an explicit isomorphism when it exists. This occurs for all unitary
symmetry groups and many groups with antiunitary symmetries, which we
characterize. We find that this isomorphism is typically not implemented by the
Jordan-Wigner transformation, nor is it a consequence of any other duality
transformation that falls within the framework of topological holography. Along
the way to this conclusion, we recover the fermionic stacking rule in terms of
G-pin partition functions, give a gauge-invariant characterization of the
twisted group cohomology invariant, and state a procedure for stacking gapped
phases in the formalism of symmetry topological field theory.
