Sometimes, but only rarely and not for very long.

Most of the people posting here don't know what they're talking about. Pedaling technique is a red herring.

Researchers were putting force-sensing pedals on bikes decades ago and looking at how people actually pedal. Even riders who *think* they're pedaling circles don't. In the late 1980's Kautz was looking at pedal forces. You can actually find his original data files here. In 1991 Coyle (and Kautz) published a paper comparing elite "national-level cyclists" with very good regional and "state-level" cyclists. Here's figure 5 from that paper:



Group 1 are the elite national-level cyclists, Group 2 are the very good state-level cyclists. As you can see, on average the Group 2 cyclists pull up very slightly on the pedal on the recovery and push down a little less on the power part of the pedal stroke. On average, the elite national-level cyclists don't pull up on the back side and stomp down harder during the power phase. That means the best cyclists have *less* "circular" pedal strokes, no matter what they thought they were doing. As for "scraping mud" across the bottom of the pedal stroke, if you examine the actual data files you'll see that they don't do that, either.

I have a friend who is a cycling coach. A few years ago he had to have the lower part of his left leg amputated. Now he concentrates on training to maximize his power and doesn't worry about pedal stroke. He can't pull up on the back side or he'd yank his prosthetic off his stump. He can't scrape mud off at the bottom of the stroke, either. All he can do is stomp down, so he stomps down hard. He has years of power files, both pre- and post-amputation. After a long period of recuperation and training, he eventually surpassed his pre-amputation threshold power. Stomping seems to work for him. [Edit:] As for one-legged pedaling, he recommends that you pedal with as many legs as you have.