States evolving in quantum circuits are known to undergo measurement induced phase transitions from volume law to area law entanglement. I argue that many more phases are possible. This statement is made precise in a broad class of circuits, for which we can map the information dynamics to the ground states of an effective Hamiltonian, possessing a higher symmetry than the symmetry of the physical circuit. The different ground states admitted by the higher effective symmetry, including broken symmetry and SPT phases, correspond to distinct patterns of information flow and scrambling in the circuit. I will illustrate these ideas with two examples. First I will show that the dynamics of spin systems having only Z2 symmetry, generally map to the ground states of an effective Hamiltonian with D4 (non abelian) symmetry. I will discuss the physical meaning of the different phases in terms of the information flow and scrambling on the circuit. Second, I will show that quadratic fermion circuits, preserving only fermion parity, map to an effective Hamiltonian with U(1) symmetry. We consequently predict a measurement induced KT transition between a critical phase with log(L) entanglement and a "trivial" area law phase.