Properly, it was a "cavity magnetron," and was the lodestone for centimetric radar development that the UK successfully developed first, but lacked the industrial capacity for quality mass production of the AI sets.
In talking with US radar technicians who serviced Beaufighters (Mk VI w/ Mk IV, later Mk VIII radars) in American night fighter squadrons (414, 415, 416 (who also flew Mosquitos), and 417), the difference in quality and standardization between US and UK produced sets was night and day.
Interestingly, the UK sets could be 'tweaked' to get either longer or shorter range contacts than the American ones.
As for inventions passed back and forth, can we call it a draw since we sent the airplane (not aeroplane) over to you?
(And stored it in London until the 1940s due to a spat between the Wright Brothers and the Smithsonian - they wouldn't totally admit the Wrights were first over Langley and the Wrights wouldn't let the original Wright flyer go to the Museum until the Museum did so.