I find it hard to explain why we turn pale when we fear but red when we are angry. In both emotions adrenaline is released which if I am not mistaken cause vasoconstriction in vessels going to the head. As such I would expect to be pale during anger instead of red. So what is causing causing the difference in color in the two emotions. It seens that in anger blood vessels dilate while in fear they constrict, I thing this contradicts that in both emotions adrenaline is released.
The 5F response (fright/flight/fight/freeze/fawn response) to threats is responsible for both situations.
Specifically, among other reactions, the 5F response causes the following: [emphasis mine]
Catecholamine hormones, such as adrenaline (epinephrine) or noradrenaline (norepinephrine), facilitate immediate physical reactions associated with a preparation for violent muscular action and :
- Acceleration of heart and lung action
- Paling or flushing, or alternating between both
- Inhibition of stomach and upper-intestinal action to the point where digestion slows down or stops
- General effect on the sphincters of the body
- Constriction of blood vessels in many parts of the body
- Liberation of metabolic energy sources (particularly fat and glycogen) for muscular action
- Dilation of blood vessels for muscles
 Henry Gleitman, Alan J. Fridlund and Daniel Reisberg (2004). Psychology (6 ed.). W. W. Norton & Company. ISBN 0-393-97767-6.
Pale face when in fear
During periods of threat, the amygdala and hyppocampus within the brain can initiate "flight" within the 5F response.
The physiological changes that occur during the fight or flight response are activated in order to give the body increased strength and speed in anticipation of fighting or running. Some of the specific physiological changes and their functions include:
- Increased blood flow to the muscles activated by diverting blood flow from other parts of the body.
- Increased blood pressure, heart rate, blood sugars, and fats in order to supply the body with extra energy.
- The blood clotting function of the body speeds up in order to prevent excessive blood loss in the event of an injury sustained during the response.
- Increased muscle tension in order to provide the body with extra speed and strength.
 Stress Management for Health Course. "The Fight Flight Response". Retrieved 19 April 2013.
 Olpin, Michael. "The Science of Stress". Weber State University.
Because the muscles in the arms and legs etc. need more blood to operate effectively, the increase in blood flow to the muscles is obtained through Vasodilation of blood vessels leading to those muscles, and Vasoconstriction of blood vessels leading to areas of the body which are in less need of blood.
Diversion of some of the blood from the facial area will remove the colour from the skin in that area.
Red face during anger
The 5F response, and resulting Vasoconstriction and Vasodilation is responsible for this too. This time with a "fight" response.
Within a few seconds, your pupils grow large, your heart beats faster, and blood vessels all over your body widen, to increase blood flow. This allows more blood to reach your muscles, providing them with more oxygen, and preparing them for action — whether that’s putting up a fight, or running away,” explains Aranda. “As a result of all this, the vessels in your face also widen, allowing more blood to flow in.
“Scientists are still learning what, if any, evolutionary function this response — known as flushing — might have, especially because humans seem to be the only animal that has it.”
“It’s likely that this redness has been useful in helping us communicate our emotions. When your face turns red, the people around you have a better idea of how you’re feeling — which helps you socially.”