The Pegasus started life as an early turbofan engine which used the Bristol Orpheus as the gas-generator. It came about because the original Orpheus had a novel design feature in that it saved the weight and cost of a centre bearing by making the mainshaft a large diameter tube whose stiffness was enough to avoid whirling without the additional support of a third bearing. Someone (Stanley Hooker, if I remember the story accurately) hit on the idea of adding what we would now call a second spool by running another shaft down the middle of this tube carrying a "big" fan at the front driven by a dedicated turbine on the back of this second shaft. This then led to thoughts of a 3-nozzle VTOL engine which had a straight-through (non vectoring) nozzle at the back, and the front fan air collected into two (vectoring) side nozzles. He drafted a brochure on this and discussed it with (again, IIRC) Ralph Hooper at Hawker. He had an idea for a light recon jet which landed vertically by pitching up to about 30 degrees so that the fixed and vectorable nozzles would cancel out to zero horizontal thrust.
At the time the Kingston project office were still hoping for a supersonic fighter project, but in the mean time they played with the VTOL concept to keep occupied. They had long since decided that any practicable VTOL jet had to be a "flat-riser" rather than any kind of "tail-sitter", so they went back to Bristol with the suggestion that the rear nozzle should be split and directed into two vectorable nozzles to produce the four-nozzle, fully-vectorable configuration we are now familiar with. They also made another suggestion which was to prove crucial. Bill Bedford had been in the US and had flown several of the US VTOL technology demostrators, including the Bell X-14 which he crashed. He crashed it because he got it into a situation where the powerful gyroscopic moments of the two engines running at high power but zero airspeed significantly exceeded the available control authority. Hawker's team had recognised that gyroscopics would be a significant issue on VTOL aircraft due to the negligible aerodynamic damping and stabilising forces in the hover. So Hawker suggested that the spools in this engine should counter-rotate, and that where possible effort should be made to try to make the gyroscopic moments of the two spools equal (but opposite).
As a result right from the very beginning the pegasus had negligible gyroscopic moments to upset hovering stability/controlability. This just left the discovery of "intake inertia moments" as the only nasty effect to be mitigated, but that's another story.
More than anyone wanted to know, I'm sure.
PDR