Steve put it in much more user-friendly terms. There are a host of other factors like a chemical mediator called 2-3 DPG that affects the dissociation between oxygen and hemoglobin and I believe that breath training effects production of it though I can cite no studies to this effect since I doubt they exist. Not enough interest, not enough test subjects. So heuristics it is for us freaks. Anyway, what matters is that it works.
With respect to cold showers, my feeling is that it is essential to start with pure cold and THEN go to hot. My experience is that the practice helps train the resiliency of the nervous system. Having hot water first makes the cold psychologically and physiologically easier to handle because one is relaxing before introducing the stress. I like to do things "cold", so that I am training to perform in response to sudden changes in events and can go from 0-100 faster since this is the way my job works. I am sleeping and then awake to a terrifyingly loud noise and then I'm racing down the street, making serious decisions with very complex variables while having to deploy both fine and gross motor skills. As a result I like to do "sub-optimal" things like training at 3 AM with no warm up, training when hungry, and cold first. I don't make "gains" like other people because of my lifestyle but I don't know as many people who can maintain as high a level of function with as little rest and nutrition as I can. I rarely get to be 100% so I learn to be "pretty good" at 70%.
Cold is a shock to the system. It triggers an involuntary hyperventilatory response, this is called quite appropriately the "cold shock" phenomenon. If you are interested in it I recommend watching my countrymate Dr. Norman Giesebrecht, aka "Dr. Popsicle's" videos on the subject of hypothermia and cold response. He is a world expert on hypothermia and does videos how to survive things like falling through the ice. I yearn for the day to see Dr. Popsicle and the Ice Man doing research and/or interviews together. I believe that one of the main, though not only, benefits of the cold training is to give the psyche a training stimulus for dealing with sudden acute stressors. I find that with repeated exposure my ability to be less "reactive" to sudden acute stressors is improved. By shocking the body and then trying to return to homeostasis in the context of the persistence of the offending stimulus, the psyche's ability to retain optimal function in the presence of a threat response is enhanced. I am less "jumpy" and less prone to sympathetic (fight or flight) tone, and when it does come on I can quickly shut it down or moderate it. This has benefits for day to day living in terms of reducing the "death by a thousand cuts" of modern life as well as high-stress situations of first response work. It also helps you understand the breath cycle. Try exhaling completely as you enter the cold. Now try inhaling as you enter the cold. See how your body responds and how your perception of the stimulus changes depending on your breath cycle. Most people instinctively inhale when they are surprised. Training to exhale gives greater psychological purchase on stressful situations.
That said, there is also a certain amount of common sense. These are stressors. Stressing the body and psyche only make sense if they are done in a way that we can fully recover from. This is ESSENTIAL to Systema work. We don't just beat ourselves up and "tough it out," we are always trying to cause stress but then take the time to properly recover from it so that we are able to fully compensate and return to normal. The notion of self-tyranny to kind of twist or numb this system is antithetical to our training methods. We use stress to stretch and soften the nervous system, not to make it rigid and brittle. Its like kneading dough.
My feeling, and that of many Systema practitioners, is that the comfort and security of modern society has left our autonomic nervous system hyper-sensitized to threats the same way that our airways are hyper-reactive to allergens (asthma, anaphylaxis.) This is particularly problematic for people who are then doing jobs where we are exposed to repeated acute stressors, but it is also a problem in the long term for chronic disease and inflammation since sympathetic activation produces an inflammatory cascade and we are slowly learning that inflammation is involved in cancer, respiratory problems, heart disease, diabetes, basically every major disease process has inflammation as a major cause.
I would say that
@pet' has provided an incomplete telling of Systema breathing practice. I speak as an Instructor-in-Training who hopes to upgrade to full Instructor in coming months. It is multifaceted. There is a principal of breath continuity and links between breath and movement are shown to be optimal BUT also a recognition of the fact that breath WILL be interrupted at times from stress (like the cold shock) and we therefore train to be effective while holding the breath, keeping the breath OUT of sync with movement, basically developing the familiarity with every breath modulation we can possibly expect. The idea is not to provide a fixed "breathing method" but to give the psyche a chance to understand and "play with" the relationship between breath, movement, perception, tension and emotion. We often intentionally adopt "box breathing" patterns involving variable timing relationships between inhale/exhale cycles and holds at either ends of these cycles. I think it is more or less identical to Buteyko. Steve is bang on with relationship between breath and relaxation. The consequences of this kind of training is that I will suddenly realized I've stopped breathing when I am particularly relaxed. I feel no compulsion to and it is rather pleasant. I suppose the reason is that my cells are being well-oxygenated and yet my chemoreceptors are very tolerant of higher CO2 levels.
@wespom9 in response to your point about sharing this stuff. Absolutely. I like to rebuke friends who complain that they wish they lived in the "samurai times" or some stupid garbage like that as if they could have been better martial artists if they'd lived in feudal Japan or been in the Japanese Imperial Army with Morihei Ueshiba. There has NEVER been a time like this. There is NO EXCUSE for the kinds of physical frailty we take for granted in this society. The information is out there. The nutrition is available. Our lifestyles are amenable to enormous free time relatively speaking. In China during the past century, one was exceedingly luck to FIND a decent martial arts master. Then came the challenge of convincing him to teach you. This required enormous expenditures of time and resources because these skills were so prized and so were not just given away willy-nilly. Now times have changed and I have had the opportunity to train with some of the best in the world, and to study the movement, writings and lectures of even more of the best. When you learned from an old master back in the day, your "learning" would be getting to watch him perform his "form" 3 times once a week. The rest was up to you. And people still did learn. What is our excuse?