  It is well-known that symmetry-protected topological (SPT) phases can be
obtained from the trivial phase by an entangler, a finite-depth unitary
operator $U$. Here, we consider obtaining the entangler from a local 'pivot'
Hamiltonian $H_{piv}$ such that $U = e^{i\pi H_{piv}}$. This perspective of
Hamiltonians pivoting between the trivial and SPT phase opens up two new
directions which we explore here. (i) Since SPT Hamiltonians and entanglers are
now on the same footing, can we iterate this process to create other
interesting states? (ii) Since entanglers are known to arise as discrete
symmetries at SPT transitions, under what conditions can this be enhanced to
$U(1)$ 'pivot' symmetry generated by $H_{piv}$? In this work we explore both of
these questions. With regard to the first, we give examples of a rich web of
dualities obtained by iteratively using an SPT model as a pivot to generate the
next one. For the second question, we derive a simple criterion guaranteeing
that the direct interpolation between the trivial and SPT Hamiltonian has a
$U(1)$ pivot symmetry. We illustrate this in a variety of examples, assuming
various forms for $H_{piv}$, including the Ising chain, and the toric code
Hamiltonian. A remarkable property of such a $U(1)$ pivot symmetry is that it
shares a mutual anomaly with the symmetry protecting the nearby SPT phase. We
discuss how such anomalous and non-onsite $U(1)$ symmetries explain the exotic
phase diagrams that can appear, including an SPT multicritical point where the
gapless ground state is given by the fixed-point toric code state.
