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Yes, they do have different part numbers, but doesn't necessarily mean that they are different in any way.
I replaced the right rear air strut today on my '91 LS400. I can say definitively that the left and right sides are *not* interchangeable, at least on the rear. The air inlet for the air strut is on the inside, aft of the centerline of the rear axle and the connection to it is made through a hole in the sheetmetal separating the trunk from the wheel well. If you put a right side strut on the left side you wouldn't be able to connect the air line.
Oddly, the Arnott website doesn't provide a way to order left or right sides when you place your order for a rear strut. I didn't know about the non-interchangeability at the time but someone from Arnott called me the next day and asked which side I needed. He confirmed that they are not the same and cannot be interchanged.
I believe that the struts are the same for '90 to '92 so it could be that later years are interchangeable. It may also be true that the front struts are interchangeable; I don't have any definitive knowledge about that.
With regard to my car, a 1993 JDM Clesior, which is the only only one I can speak for - the FRONT shocks are interchangeable side to side but the REAR are not.