CR is a micromanager's ultimate dream tool. You can use it to switch civics several times in one turn. This leads to the concepts of instant civic groups and end-of-turn civic groups.
Instant civics are used to:
a) rush buy some buildings or units using just US to inject hammers into selected cities.
b) draft troops with Theocracy giving a promotion
c) diplomatic manouvers switching civics and religions (although positive aspects take time to build negative aspects are removed immediately). Once arranged they stay in place for several turns.
End of Turn civics are those that affect production, research and GPPs.
Production bonusses from normal hammers, whipping and chopping for units (Police State +25% Military units) and buildings (Organised Religion) can effectively give more hammers.
US gives hammers with towns
Free Speech gives +2 commerce with towns
Vassalage (+2 exp) and Theocracy (+2 exp) give better quality units
Nationhood gives +25% EPs
Pacifism gives +100% GPPs
Mercantilism, Free Trade, State Property and Environmentalism affect trade routes and several other important game effects.
Obviously only one set of End of Turn civics can be used but that doesn't have to be the same set every turn and different cities can be doing several things at once on successive turns. For instance; one turn a city could be working maximum food tiles to fill its granary (and not care much what the civics are) and the next turn it could maximise the number of merchants it runs, starving the city severely, but the civics could be Caste System, Pacifism and perhaps Representation. Several cities could be doing that to make GPs (about twice as fast for a non philosophical civ) for another Golden Age or for a corporation, while other cities could be cranking out military units with good experience on alternate turns or building infrastructure.
I could go on

detailing many other combinations but I'll let you think of them yourself. Needless to say it is one of my favourite new wonders and rivals the Mausoleum in terms of its game altering ability.