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Sway bars are not really about body roll. They are about tuning the stability of the car. More stiffness in front will increase understeer, more stiffness in back will increase oversteer. What causes the sway you are talking about? Is this connected with cornering or maneuvering? Worn out struts will cause overshoot on body response, which you would perceive as sway.

As far as straight line stability, alignment is critical, and that is the most common problem. Also consider the aerodynamic issue mentioned above.
 

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... a staggered build is generally for performance vehicles. How does the Spyder qualify as a performance vehicle, or perhaps, why didn't the Toyota engineers come up with a square build for Spyder?
In this case, the staggered tires are to compensate for the rear weight bias. Rear heavy cars are intrinsically unstable, and that needs to be compensated to make the car safe and drivable. There are other means of compensation: Tire pressures, alignment, sway bars all can contribute, but in my opinion, matching the tires to the load is the most natural, and the least likely to cause side effects. Bigger feet, bigger shoes. Some drivers like an unstable car, and some drivers are not put off by the rear end stepping out quickly. However, if spinning the car is not in your plans, I would recommend keeping the 2-3 cm stagger that Toyota fixed on. I had a mk1 for 30 years with the square tire setup, and the spyder is a vastly better handling car, particularly near the adhesion limit.
 

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It is the rear weight bias that makes the car easy to spin. The low moment of inertia actually makes the car easier to recover from the adhesion limit. The car will respond more quickly, and it will require less lateral force from the tires to get the same response.

The tires have to produce lateral force for two different degrees of freedom. There is lateral acceleration, or cornering, and there is also angular acceleration, which is turning in and straightening out. During these two different stages of cornering, the forces add and subtract differently. Now if you consider a rear heavy car with the rear tires overloaded, you can figure out why it is easy to overcommit in a corner and why the car will spin at unexpected times. That is why you don't want to overload the rear tires unless you know what you are doing and can plan your line far enough ahead.

The three secrets of spyder handling are the long wheelbase, the low moment of inertia, and the staggered tires. The wheelbase is often overlooked, even though it is equally important to the moment of inertia. The spyder has the longest wheelbase of the mr2 series, even though it is shortest overall. The staggered tires are what allow you to snap the car straight after a corner, without the rear end plowing.
 

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Spoilers work. It is the wings that have questionable utility. There is popular confusion as to which is which, and there are a lot of purely cosmetic appendages around that add to the confusion.
 

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A wing provides downforce. A spoiler interrupts airflow for the purpose of controlling turbulence. That is the origin of the name "spoiler." A spoiler on the rear deck is usually to reduce drag. A spoiler in a different location is probably very specific to the airflow at that point. A fairing or air dam is just supposed to exclude the airflow from a certain space.
 
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