There are some dissenting opinions about the value of spinning. They state that you cannot add any significant energy to the pedals by pulling up with the other leg.
I don't know whether I buy that, but I have concluded that pulling up with the other leg at least gets the weight off of the pedals while the power leg is doing the work.
But, one of the better things that I have read about spinning properly (and it comes from Greg Lemond, by the way) is to think about pulling the knee forward toward the bars (rather than pulling the leg up toward the sky).
All that said, I agree that a good cadence is one of the most important aspects. As is just tryng to make sure that your pedaling action is smooth. I rode with someone recently who had a set of goofy pedaling techniques. She was most annoying to ride behind because her bike was all over the road, just from her pedal action alone.