skribs
Grandmaster
You linked sport to symmetric and street to asymmetric in your next paragraph.
Maybe I didn't make it as clear in my OP as I thought I did, because @gpseymour and @Tony Dismukes both read it the same way you did. Let me try and rephrase it:
There are drills, and there are arts. Symmetric arts will use symmetric and asymmetric drills. Asymmetric arts will also use both, but probably lean more heavily towards asymmetric drills.
It's also not a perfect definition, just like any attempt to categorize all martial arts isn't going to be a perfect definition. The Taekwondo we do at my school, we have some things that are solo or performance-based, some things that are symmetrical, some things that are asymmetrical, in their end goal. We have self-defense training, which is asymmetrical in nature. We have sparring, which we go to tournaments, which has a symmetrical goal.
My goal of this post isn't to tell everyone how to think about arts. It's to provide another way of looking at what you train, from a different perspective. Sometimes that different perspective gives you new insight, sometimes that new perspective is completely useless.
At the very least, it gave me a different way of thinking about things, so I'm happy with it.