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.