The main functionality for ordering gates in a cirq circuit that I'm aware of is using the predefined insert strategies of
cirq.Circuit.insert(). As far as I see, none of the predefined strategies use the possibility to shuffle commuting gates to reduce circuit depth, i.e. that CNOTs that share the same target or control commute, etc. The docs also don't seem to mention anything like this.
I'd like to specifically use the commutation rules of CNOTs to reduce the depth of my circuits, and before I start writing my own code, my question is: does cirq provide any functionality of this sort?