Cycling with mirrors, modulo n

For an integer n, we wish to dispose the n-1 non-zero elements of
Z_n in a circular arrangement [a_1, a_2, ... , a_{n-1}] (a_n = a_1)
so that the set of differences a_{i+1} - a_i (i = 1, 2, ... , n-1)
is itself Z_n \ {0} .   A basic recipe for doing this for any odd
n  was given in 1978 by Friedlander, Gordon and Miller.   Variants
of this recipe are available for  n \equiv 1  (mod 2i), with
i = 2, 3, 4, ... .    Entirely different procedures have been
published by the speaker for prime values of  n .   Now procedures
have been found specifically for values  n = pq  where  p  and  q
are distinct odd primes.   Many of these new procedures produce
arrangements in which any element  x  and its negative  -x  (mod n)
are  (n-1)/2  positions apart (in either direction).

The talk will pay little heed to possible applications.   However,
for small values of  n , the arrangements give us neighbour designs,
as in statistical design theory.