On the quartic surface $x^4+y^4=z^4+w^4$

Swinnerton-Dyer (1969) states a remarkable theorem that describes all curves of arithmetic genus 0 (hence parametrizable) on the quartic surface of the title; but apparently he never published or gave any details of the proof. Here, we flesh out his skeleton, and in consequence can now give an explicit description of all such parametrizations of degree up to any preassigned bound. In particular, it turns out there are 86 distinct such (non-trivial) parametrizations of degree less than 50.