Answer: The short answer is that in a 3rd species exercise (4 notes against 1 of the cantus) a cambiata must begin on an octave if it is written above the cantus, and it must begin at the interval of a 3rd (or 10th, etc.) if written below the cantus. This is because the classical cambiata consists of 4 notes in which the 1st, 3rd and 4th notes are consonant. If you try a few experiments you'll see that a cambiata above a given bass must start on the octave to meet those requirements. And when written below it must start with a third (or third plus octaves, i.e. 10ths: same thing really). The classical cambiata is a second down to a dissonant tone, then a downward leap of a third, followed by an upward second. The 4th note is like a delayed resolution of the dissonance in the second note.

In free counterpoint the possibilities multiply because the bass doesn't have to stay the same; it can move to a different pitch before the cambiata finishes. So in free counterpoint a cambiata can begin on any consonance.