Thermostatic valve operation; is the hot or the cold or both controlled?

Does a thermostatic valve control the hot flow and let the cold flow freely, or does it control the cold flow and let the hot flow freely? Or does it close one off as it opens the other?

And while I have the attention of you thermostatic experts; How fast is a "fast" responding thermostatic valve? Several seconds?, about 1 second?, small fraction of a second? Do all thermostatic valves also incorporate a pressure balance to handle the sudden changes while the thermostat slowly adjusts for the long term temperature?

Usually it will control the hot water. If the temperature starts rising it will shut down the hot water, and if there is not enough cold water to balance it, it will eventually shut the hot water off entirely. If there is no hot water, it would try to keep adding more, but the water would just stay cold. Most have a paraffin "motor" and it operates fairly rapidly, but NOT instantaneously. Most do not have "room" for both pressure balance and thermostats, and a pressure balancing feature would be redundant.