Does anyone know why step 3 of the natural zig zag from PP in foxtrot (howard) is facing DC for the lady but backing centre for the man ?

Interested.I would say the reason is, done any other way the man would be standing exactly where the lady is supposed to go. If the man was Backing Diag Centre what would the ladies alignment be.
Another good reason we can give you is because the book says so.

Suggested Answer: Facing DC. In the same way that in a back lock the man's alignment is facing DW whilst the lady's alignment is backing DW. In the case of the back lock, there is not a problem of the kind you suggest in having exact opposite alignments (as that is taken care of by having the lady step OP), so why should such a probem arise with the natural zig-zag ?

Related question (answer not in the book): What would the man's alignment be for step 3 of a natural zig zag commencing not in PP but with lady in line, man facing, lady backing DW ?

Re: variations that aren't in the book. I have a few more questions, and wonder if anyone could suggest answers or point out the factors involved.

(1) Open Impetus turn on normal alignment followed by Chasse from PP moving DC. What are the foot positions/aligments on step 3 of the open impetus and step 1 of the Chasse from PP ?

(2) Open Impetus Turn commencing backing DW, followed by the Natural Weave. What would the foot positions/alignments be on step 3 of the open impetus and step 1 of the natural weave ?

(3) Open Telemark followed by natural weave. What would the foot positions/alignments be on step 3 of the open telemark and step 1 of the natural weave ?

Interested. It might be worth looking at the fine print on both the Open Telemark as well as the Open Impetus. You will find there are different degrees of turn In one case less for the man and more for the lady. Which does have an effect on the finishing position which is, for both couples going into Promenade, should be looking on the outside of their hand and not the inside.
What i was told years ago is that a mistake the lady makes on an Open Telemark is they see where they need to go and stop turning when actually there is still a little further to go.
Let us not forget that Ballroom dancing is not a Science it is an Art.

Actually, it is a Science based on a set of scientific principles ( Kinetic energy, Dynamics etc )... we hope, and try, to translate those into an Art form..

Ballroom Dancing a Science. It just doesn't sound right. The music we dance to, can that be called a science also. That also doesn't sound correct.
Anything that is open to interpretation meaning two couples doing the same moves and yet both have a different interpretation. Then that is an art. The Art Of Ballroom Dancing.

I agree entirely with the thinking expressed by terence2. Artistry depends to some extent on precision and in these cases on geometric precision (ie science based).

The problem posed by my original questions come entirely from the differential alignments on coming into promenade that you point out, which are presumably built in to ensure the both partners feet end up on the same coordinate along the line of movement.

Any answers to my original questions still graciously received.

Also one more scenario I thought of. If you want to do a chair line after an open impetus would it be forward & across in CBMP & PP or just fwd in CBMP & PP ? The latter feels insufficiently balanced for this figure because of the RF moving further outside of the body, and also, because of the more open foot position, the extent to which both knees can be flexed simultaneousy is more limited anyway. But the former will cause a mismatch with your partners foot position. Maybe some combinations can never work out perfectly.

Interested. The questions you are asking you could ask on any move any dance. Why stop at a Chair. You should do it in front of your coach and let them decide if it can be improved on.
You might ask yourself when you step across your body whilst in Promenade exactly where is your body facing. And also what do you intend to do on the completion of the Chair Line.
Basic Footwork which doesn't alter no matter what we do. Slide your foot forward first on the ball and then as the heel passes the toe of the supporting let it become a heel at the same time start to raise the heel of the supporting foot..
Back to the Chair. Do the CBMP part correctly with the above footwork on your entry into the figure and you wont be far wrong. Let your coach do the do the rest. It is our belief that if we did a prolonged Bronze medal Class and not one of those short throw you in at the deep end classes. Having learnt the Basics thoroughly we would finish up much better dancers. And we would know exactly the correct footwork and alignment when we do get taught more advanced things like the Chair.

Here is where the technique book is Confused. The Double Reverse Spin the suggested best alignment is to commence down the LOD which requires one full turn to finish facing the LOD.
And yet the figure suggested to follow is an Open Telemark. This is supposed to start facing diag to centre. This would require more than a full turn and could become very untidy for the average dancer. Google Mirko and Alessia the Blackpool Champions and see what alignments he uses for a Turning Lock after an Overturned Spin Turn. You will find that the first three of the Natural Spin Turn is under turned. Which confirms, in our opinion, that Ballroom Dancing is an Art and not a Science.

I suspect that the confusion lies elswehere.

The range of turn for the DRS is 3/4, 7/8 or 1 complete turn. Howard, and others, chart 7/8 as being the standard turn, and an Open Telemark would nicely follow a DRS commenced facing LOD, ended facing DC (although that is not one of the suggested follows).

Alex Moore mentions the Telemark follow facing LOD after a complete turn in "Ballroom Dancing", but in the later "Revised Technique" he adopts the normal convention. Presumably even a Blues dancer can learn from his peers.