In an open differential, while the tires are off of the ground, why does one wheel turn backwards when I turn the other wheel forward?

When turning a corner, the inside wheels must turn slower than the outside wheels, because the outside wheels are having to cover a longer distance, on the outside of the arc of the turn, than the inside wheels do on the inside of the arc of the turn. (Huh??? I will have a picture soon; it might explain this better than I am able....).
Now, it stands to reason, that if we hooked a speedometer up to one drive wheel, and another speedometer to the opposite drive wheel, (on the other side of the vehicle), and yet a third speedometer to the ringgear in the final drive (rear end, in a rear wheel drive vehicle), that all three speedometer should read the same when we are travelling down the highway in a straight line (if they do not, then the speedometers are not calibrated the same, or the drive wheels are two different diameters....). OK, so they are all reading the same. Now, if we turn a corner, the inner wheel must turn at a slower rate than the outer wheel (hopefully understood above....), so the inner speedometer will read slower than the outer speedometer.... right? Here's where it gets fun to explain (at least for me... I am not very good at this!). Let us say that the inner speedometer happens to be reading 38MPH, and the outer wheel happens to be reading 40MPH (I have not run the numbers, so I cannot tell you just how sharp a turn this is, but let us use these numbers to, hopefully, illustrate my point (there is a point here??)). The ringgear is driven, indirectly, by the engine, and transmission, and is turning at wheel speed when cruising in a straight line. The axles are connected to the ringgear, through the differential, which has some gears in it(they are called side gears, and spider gears). The differential's job is to transfer turning motion, from the engine, to the axles, but allow for the speed difference of the two axles, again, when turning a corner (am I rambling here, or what???). The way the differential accomplishes this task is by speeding up the outer wheel's axle, to something faster than ringgear speed, and slowing down the inner wheel's axle to the same amount, slower than ringgear speed. Now, imagine, for a moment, that the ringgear is stopped, but the outer axle is still turning faster than ringgear speed, and the inner axle is still turning slower than ringgear speed. For the moment, when the ringgear is stopped, the inner axle is turning slower than ringgear speed (which is completely stopped), so it must be turning backwards, and the outer axle is turning faster than ringgear speed, so it must be turning forward. The point??? OH YES!!! The point!!! This is why, when the vehicle is stopped, and jacked up on your driveway, one rear wheel turns forward while the other rear wheel turns backward. One interesting thing to note: In the example of the inside wheel turning at 38MPH and the outside wheel turning at 40MPH, the ringgear speedometer would be turning at 39MPH (the average of the other two speeds). Also note that if one wheel gets no traction, all of the turning torque will be transmitted to that wheel with no traction. So, when the speedometer, on the ringgear, reads 40MPH, the wheel with no traction will be turning 80MPH, and the wheel with traction will be turning at 0MPH. When this happens, your stuck.

Figuring Gear ratios

If you ever want to figure out your gear ratio, you just jack up one wheel, and turn it two (not one, like you may have thought) complete turns, while you count the drive shaft rotations. Example: if, while turning the wheel two complete turns, the driveshaft turns a little more than 4 turns, you may have a 4.11 gear set in your vehicle. Of course, the best way is to count the teeth on the ringgear, and the pinion, to get the most accurate reading. Example: 37tooth ringgear, and a 9tooth pinion gear will yeild 4.11.

Positraction, or Limited Slip

Limited slip rearends work in exactly the same way, but, they resist this "turning at independant speeds" idea, by installing a set of clutches (or something else that may act like a clutch) on one, or both axles, and "hooking" the axle, or axles, to the ringgear carrier. The clutches are set to slip at a certain torque. By design, as input torque, from your engine, increases, the clutches hold on tighter, in an attempt to keep both axles turning at the same speed, relative to each other. When one wheel has no traction, a limited amount of torque is applied to the opposite axle, hopefully helping enough to get you "unstuck". And the difference between "Posi", "Limited Slip", "Traction LOK", etc? They have all become generic overused terms that basically can be used interchangeably. "Positraction" is a Trademark name, just as "Traction LOK", but have become generic terms, just as "Cresent Wrench", and "Channellocks", although both being specific brands, have become generic terms. ("CresentWrench" is a particular brand of adjustable wrench, and "Channellock" is a particular brand of adjustable slipjoint pliers). Another example is the term "COKE", which many people use as a generic term meaning Cola, even though "COKE" is a particular brand of Cola (Although the author prefers Pepsi, but definately not because they chose Micheal Jackson, or the Village People, as representatives.... Definately NOT!! ) .

"Lockers"

Lockers are a variation on the traditional differential, but do not work anything like a conventional differential. While a locking differential must also allow for different wheel speeds while cornering, they don't like to, nearly as much. They do, however, lock up, on straight pulls, not allowing for any slippage, at all, so when one wheel loses traction, the other wheel gets all the torque the engine can deliver. This can be good, and bad, depending upon how you look at it. I like lockers for rocks, but you have to be careful; if you are hard on the throttle, and one tire breaks traction, the other axle gets 100% of the torque created by the engine, which can lead to breakage. In addition, since they do not like to unlock in high torque conditions, cornering can be difficult, if engine torque is high. Also, they can be very hard on a vehicle, and it's axles, that is used on the street. I would not recommend using a locker on the street, unless you use your vehicle mostly off road, and only rarely use it on the street. The exception is the ARB (Air Locker); it remains unlocked until you "turn it on" with air pressure, where it locks until you turn it off. The drawback, I am told, with the ARB is that seals do not last and the ARB requires quite a bit of maintenance. Among the others available, are the Detroit Locker, and the Loc-rite.