What he said, but I turn the wheel the other way. So, my right foot is forward, and I turn the wheel to the left and use the crown in the road as the hill (usually). There's a web site on fixed gear riding that gives a much better explanation. With the wheel turned, moving forward corrects one way, moving back corrects the other. Fixed gear people can do this w/o a hill. Geared or SS people can do this using the brakes instead of a hill as you suggest. The idea (in my mind) is that by shifting your weight quickly forward to back as you release the brake creates the backwards movement, but this is subtle. It is much easier to learn w/ hills.
I just started standing for as long as possible on my commute, and a few months later was rarely putting a foot down over full light cycles. And, it's easier on my touring bike w/ 27" wheels than on my mtb.
Don't know why.