Question about Shofar Technique

After this year’s set of RH services, I have some questions about the shofar. I was told that it’s traditional to play it on the right side of the mouth*. This year at my, shul, we had three different shofar blowers. 2 played on the right side of the mouth, one in the centre. Why is the right side significant? Does it change based on handedness? Is it okay to play in the centre?

I am, by no means, a shofar blower myself, but I did have that role for 2 years at my college Hillel because we didn’t have anyone official to do it, and I play trumpet, which means I can get a pretty nice sound out of a shofar without having been taught a specific shofar technique. One thing that bugged me was that I was never able to play it from the right side. I only had enough control in the middle to get a clear, consistent tone, because that’s how trumpet embouchure works. So what’s different about a shofar? Why do we do it this way?