  Symmetry is a powerful tool for understanding phases of matter in
equilibrium. Quantum circuits with measurements have recently emerged as a
platform for novel states of matter intrinsically out of equilibrium. Can
symmetry be used as an organizing principle for these novel states, their
phases and phase transitions? In this work, we give an affirmative answer to
this question in a simple adaptive monitored circuit, which hosts an ordering
transition in addition to a separate entanglement transition, upon tuning a
single parameter. Starting from a symmetry-breaking initial state, depending on
the tuning parameter, the steady state could (i) remain symmetry-broken, (ii)
exhibit the average symmetry in the ensemble of trajectories, or (iii) exhibit
the exact symmetry for each trajectory. The ordering transition is mapped to
the transition in a classical majority vote model, described by the Ising
universality class, while the entanglement transition lies in the percolation
class. Numerical simulations are further presented to support the analytical
understandings.
