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Without load, both axles of an open differential will turn together.
Look at the female splines on each side of the differential assembly. The splines on one side of a LSD are in two pieces, one inboard of the other. One of those is connected to the clutch carrier.
Here's another test: Try holding both the housing and one axle steady, and turn the other axle. If you can easily spin it and the pinion, it's either open or has really worn clutches. Ford's standard spec for the preload torque on the clutches is about 40 lbf-ft.
If the AXLE code stamped on the Warranty Plate (1959 thru 1980), printed on the Certification Label (1981 thru today) begins with a letter: Limited Slip.
If the AXLE code begins with a number: no Limited Slip.
In 1935, Auburn introduced the two speed Columbia rear end. Auburn and Columbia were owned by Errat Lobban Cord, who also owned Lycoming and Duesenberg. The FWD Cord L-29 was available 1929/32, 'Coffin nosed' Cord 810/812 in 1936/37.
Ford offered the Columbia as an option in 1940/48 Passenger Cars. This unit is vacuum operated. hit the dash mounted switch, a screech is heard from the rear end when it changes gears.
Auburn LSDs use a cone-type clutch, so it's going to look different from Ford's Traction-Lok. I've never actually seen one apart, so I don't know what it looks like. However, I know that once its clutch wears out, it's not a simple matter of replacing clutch plates as in the Fords. I think the entire differential assembly needs to be replaced, or you have just an open differential.
And I agree with Bill; I've seen a lot of people incorrectly refer to Ford's LSDs as "Posi". Brainwashed by the mighty GM corp.