I agree, cold shower has a lot of benefits. With regards to Vascular Eagle Syndrome, I hypothesize that the relief comes from the vascular constriction that the cold induces on the blood vessels particularly the Jugular vien and the collateral Veins & vertebral venous plexus which get blood rerouted to them due to the compression of jugular veins. Since the jugular vein is compressed the blood tend to backup to the brain and the collateral Veins & vertebral venous plexus, so when a cold is introduced to the body, Veins constrict forcing the venous valves to open to push blood out relieving the congested veins around the neck. The other mechanism that I hypothesize is, since all vessels are constricted when in cold shower, the net blood flow is reduced, thereby decreasing the intercranial pressure a bit.
I have experienced this myself. if I go out in the winter with little clothing, My neck pain and tightness along with head pressure is relieved. So I was wondering how can that be. Though it is just my thought, the mechanism of feeling relief had to do with vasoconstriction in part but this theory needs to be tested. The cold acts like compression socks that help with varicose vein disease and push blood up to the heart by opening the superficial venous valves .
Another important feature of cold shower is the increase of the mitochondria (the cell power producers). All the high energy demanders such as the brain and muscles use it and a decline of mitochondria is linked to a number of neurodegenerative diseases and premature aging (The effects of exercise and cold exposure on mitochondrial biogenesis in skeletal muscle and white adipose tissue), ( Mitochondrial Dysfunction and Chronic Disease: Treatment With Natural Supplements) ( The Mitochondrial Basis of Aging)