Understanding orthogonal regression
Nice question.

Short answer: the distinction between independent and dependent variables may not matter to your loss/error function, but it might matter for your modeling function.

Think about fitting a parabola with orthogonal regression.  Here, it's really important which variable is dependent and independent, because the function is not one-to-one.  So I'd say the distinction matters more for "modeling." 

In a simple case, such as fitting a line, then any distinction is probably more philosophical than practical.