Why Did the Early English Saxons All Call Themselves English?

When the Anglo-Saxons started to arrive in the 4th and 5th centuries the Saxons were a much more powerful tribe in northern Europe. So there was a natural tendency to refer to all of these earlier invaders as Saxons. But this raises an interesting question, why did the early English Saxons all call themselves English? What did they call themselves Saxons? After all that was what they were called by everyone else.