The "pedaling squares" comments usually or exclusively refer to climbing. While on that subject, here's a tip: a very fast way to learn just how efficiently you can pedal is to bonk a long way from home.

Seriously, pedaling in circles is what everyone does who rides a bike. Doing it efficiently isn't a branch of the mystical arts. (Remember, too, that when riders in the 1960s were telling each other to pedal circles, sports coaches were telling their players to train themselves to work efficiently without drinking water and to take salt tablets to replenish the salt they were losing through sweat.)

Getting good at pedaling doesn't require rollers or drills. Race your bike for a season or two, learn to keep up with faster riders for hours on end, and you'll become pretty much as efficient as Jacques Anquetil.

He was the rider held up as the exemplar of smooth pedaling when I started racing in 1964 (in road races, on a track bike; legal back then). Of course, he wasn't really a smoother rider than his peers. Thin riders always look sleeker and more elegant, and he was all those things.

Ironically, on being asked why he was so good at time trialing, the explanation that he gave ran counter to the "pedal in circles" mantra: he said that he always counted his pedal strokes such that he pushed once every three strokes and rested on the other two. When I read that, I stopped worrying about pedaling smoothness and started counting: RIGHT-left-right-LEFT-right-left, etc. That worked for me. I was never a cat 1, but I won a few races and set a 10-mile time trial course record that stood for a while.

By the way, I'm surprised that no one in this thread has waxed rhapsodic on the glories of "ankling," which is another thing that some veteran riders used to tout.