I have had a similar experience, albeit while skydiving. It seemed as though every time I tried to breathe, a lot of air tried to rush into me and it was very uncomfortable to the point where I didn't want to breathe. When I've biked really fast down from the top of a very tall hill, I've noticed a similar phenomenon.
From an aerodynamic perspective, I believe it is related to the build-up of dynamic pressure and stagnation pressure. When moving through the air at a fast enough velocity, dynamic pressure builds up on the front surface of an object because air is moving fast with respect to the object itself. Additionally, because of the compressibility of air, pressure also builds because the air itself is being slowed down roughly adiabatically. Think of it as though the air molecules are slowed down and squished more tightly together in front of your mouth.
Because the overall pressure is higher in front of your mouth than in normal static air conditions, there is a larger difference in pressure between it and your lungs. Whenever there is a larger pressure gradient, there is a very fast flow of air. My guess is that the discomfort we experience as extreme sport enthusiasts is related to the fact that our lungs usually only intake air quickly when we exert a lot of effort in our diaphragm muscles to expand our lungs (i.e. lowering the pressure in our lungs to allow air to flow in them). If a lot of air tries to flow into our lungs without this muscular effort, it is very opposite of our natural experiences. I postulate that our bodies, as a safety mechanism, react to this as a bad thing and our bodies naturally want to prevent that from happening. Our bodies may be reacting by breathing less to prevent undesirably large influxes of air. It makes it hard for you to control your diaphragm properly to let in the right amount of air at the right speed.
This explains why you would find it easier to breath when you turn your head to the side. Air pressure is not as high off to the sides rather than directly against the wind. The air is still moving fast, so dynamic pressure is still a little higher. But the air is not being compressed and slowed down as much as it is when your head is directly against the wind. I too found that tilting my head upward while skydiving helps me breath a lot more easily.