  We study a one-dimensional (1d) system that shows many analogies to proposed
two-dimensional (2d) deconfined quantum critical points (DQCP). Our system is a
translationally invariant spin-1/2 chain with on-site $Z_2 \times Z_2$ symmetry
and time reversal symmetry. It undergoes a direct continuous transition from a
ferromagnet (FM), where one of the $Z_2$ symmetries and the time reversal are
broken, to a valence bond solid (VBS), where all on-site symmetries are
restored while the translation symmetry is broken. The other $Z_2$ symmetry
remains unbroken throughout, but its presence is crucial for both the direct
transition (via specific Berry phase effect on topological defects, also
related to a Lieb-Schultz-Mattis-like theorem) and the precise characterization
of the VBS phase (which has crystalline-SPT-like property). The transition has
a description in terms of either two domain wall species that "fractionalize"
the VBS order parameter or in terms of partons that "fractionalize" the FM
order parameter, with each picture having its own $Z_2$ gauge structure. The
two descriptions are dual to each other and, at long wavelengths, take the form
of a self-dual \emph{gauged} Ashkin-Teller model, reminiscent of the self-dual
easy-plane non-compact CP$^1$ model that arises in the description of the 2d
easy-plane DQCP. We also find an exact reformulation of the transition that
leads to a simple field theory description that explicitly unifies the FM and
VBS order parameters; this reformulation can be interpreted as a new parton
approach that does not attempt to fractionalize either of the two order
parameters but instead encodes them in instantons. Besides providing explicit
realizations of many ideas proposed in the context of the 2d DQCP, here in the
simpler and fully tractable 1d setting with continuous transition, our study
also suggests possible new line of approach to the 2d DQCP.
