I think I may have an explanation for the left right piano issue. First, it is clear that the piano was really on the left (Joel's other sources) or on the right (the TP tape) of the performance space with respect to the listener during the recording session. Second, according to the liner notes in TP-009, Mose Allison is singing and playing the piano. Third, Mose Allison's voice is approximately in the center of the sound field. (I am assuming that is true in Joel's other sources also). Fourth, unless there are photos from the recording session or we have some documentation that the master tape has channels reversed, we may never absolutely know what happened. Paul can confirm whether the master tape channels correspond to the TP-009 channels. My guess is that Joel's sources probably are not in a position to make that same confirmation. However - now for some forensic channel sleuthing.
Here is what I think - inspired last night while watching Diana Krall in her concert in Rio on bluray. If the piano is a grand, then the player always sits to the audience left of the piano, since the lid of the piano opens forward only if the pianist is to the left of the piano. Since Allison is singing and playing the piano at the same time, and Allison is in the middle of the stage (soundfield), it stands to reason that the piano has to be on the right of the performance space. If it were on the left, then Allison's voice would be coming from the far left.
That is exactly where Diana Krall sits, playing the piano and singing - the piano sounds to right of center and her voice is pretty much centered. Her guitarist is sitting on the left, and his sound comes from the left, with drummer at left center and bass player at right center. behind Krall.
It is possible that Allison is playing an upright at the left center of the performance space. If that is true, then the piano and Allison's voice should be in almost the same location - since the upright is very close the performer.
BTW, occasionally, the piano is facing directly back from the audience (as in Barenboim's Beethoven Concerti bluray, where he is conducting and playing with back to the audience). However, that wouldn't happen with a vocalist/pianist, since he/she would be singing with back to the ausience.
Larry