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I've noticed that quite often I cannot get a single gate going AWL. But nearly always can get one going AWR. I've also read things to that effect in various threads and in The National Locksmith's Guide to Manipulation.

Does anyone have an explanation for this phenomenon?

Two reasons come to my mind why there are differences. The first is that wheel positions vary with the rotational direction. If L50 and R50 especially for W1 is not the same position. If L50 is defined to be 50, then R50 is something like 47 or 48. This can make a difference is shadowing. The second reason is wheel float, which is very noticeable in the La Gard locks. The wheels sit rather loosely on the wheels post and a change of rotation creates a change in the height profile of a wheel. Don't know why you have more success with AWR. Either it is pure chance or perhaps this lock has another effect, which lifts W3 in all cases when turned right.

Thanks for your response.

This is true of both my Big Red Wheel CDL-3 and my S&G 6730. However, my 6730 has very little offset when going in either direction. I can open the lock without a problem by putting in the combination LRL or RLR. I also checked the offset by doing AWL to 50, then turning Right to pick up each wheel. I picked up each one almost spot on 50. I got the same results from AWR to 50 as well. I do notice that on the 6730, W3 gets shadowed from around 70 until at least 0.

Your explanation of change of height depending upon direction tends to make sense. I don't know if this is the reason, but it sounds plausible. Very interesting.

A nearly perfect 50-50 is very rare. My old Mosler friction roller fence lock has that and the current S&G vault locks. My R6730 has a very noticeable offset. So it seems they did not only change the gate width.