if you're using a standard least-squares fitting to find your regressions, the linear fit that you get by minimizing the sum of squared errors in one variable is not the same as the linear fit that you get by minimizing the sum of squared errors in the other variable.

Technically speaking, if both your variables (x AND y) have errors in them, the ordinary least-squares regression is the wrong methodology to use. See http://en.wikipedia.org/wiki/Errors-in-variables_models