I think generally nowadays I swing fairly naturally (just my left arm cos I tuck my handbag under my right one), but I know I have had times of sort of having to think through really 'natural' things like breathing and swinging. When I was a kid, I just to sort of march with my legs, arms, elbows and head all jerking around!
It is thought by some people that a correct cross-pattern (arms and legs swinging on the opposite side if that makes sense) is essential to development, and that if a correct pattern is not established you will suffer from brain damage, thus I spent almost a year crawling, running and creeping to get a correct cross-pattern when I was 10 years old! Scarily, it helped! And just for the record, it's not just about balance, as not all animals use the cross-pattern.
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...though fire may burn & flames envelop me, I will arise from the ashes...