  We generalize the hidden symmetry-breaking picture of symmetry-protected
topological (SPT) order developed by Kennedy and Tasaki in the context of the
Haldane phase. Our generalization applies to a wide class of SPT phases in
one-dimensional spin chains, protected by an on-site representation of a finite
abelian group. This generalization takes the form of a non-local unitary map
that relates local symmetry-respecting Hamiltonians in an SPT phase to local
Hamiltonians in a symmetry-broken phase. Using this unitary, we establish a
relation between the two-point correlation functions that characterize fully
symmetry-broken phases with the string-order correlation functions that
characterise the SPT phases, therefore establishing the perspective in these
systems that SPT phases are characterised by hidden symmetry-breaking. Our
generalization is also applied to systems with continuous symmetries, including
SO(2k+1) and SU(k).
