As a neuroscientist, despite my initial incredulity, I came to realize that yoga works not because the poses are relaxing, but because they are stressful. It is your attempts to remain calm during this stress that create yoga's greatest neurobiological benefit.
Your brain tends to react to discomfort and disorientation in an automatic way, by triggering the physiological stress response and activating anxious neural chatter between the prefrontal cortex and the more emotional limbic system. The stress response itself increases the likelihood of anxious thoughts, like "Oh god, I'm going to pull something," or "I can't hold this pushup any longer". And in fact, your anxious thoughts themselves further exacerbate the stress response.
Interestingly, despite all the types of stressful situations a person can be in (standing on your head, running away from a lion, finishing those TPS reports by 5 o'clock) the nervous system has just one stress response. The specific thoughts you have may differ, but the brain regions involved, and the physiological response will be the same. The physiological stress response means an increase in heart rate, breathing rate, muscle tension and elevation of cortisol and other stress hormones.
The fascinating thing about the mind-body interaction is that it works both ways. For example, if you're stressed, your muscles will tense (preparing to run away from a lion), and this will lead to more negative thinking. Relaxing those muscles, particularly the facial muscles, will push the brain in the other direction, away from stress, and toward more relaxed thoughts. Similarly, under stress, your breathing rate increases. Slowing down your breathing pushes the brain away from the stress response, and again toward more relaxed thinking.
So how does this all fit together? As I stated before, the stress response in the nervous system is triggered reflexively by discomfort and disorientation. The twisting of your spine, the lactic acid building up in your straining muscles, the uneasy feeling of being upside down, the inability to breathe, are all different forms of discomfort and disorientation, and tend to lead reflexively to anxious thinking and activation of the stress response in the entire nervous system. However, just because this response is automatic, does not mean it is necessary. It is, in fact, just a habit of the brain. One of the main purposes of yoga is to retrain this habit so that your brain stops automatically invoking the stress response.