  We define a new congruence relation on the set of integers, leading to a
group similar to the multiplicative group of integers modulo $n$. It makes use
of a symmetry almost omnipresent in modular multiplications and halves the
number of residue classes. Using it, we are able to give an elegant description
of some results due to Carl Schick, others are reduced to well-known theorems
from algebra and number theory. Many concepts from number theory such as
quadratic residues and primitive roots are equally applicable. It brings
noticeable advantages in studying powers of odd primes, and in particular when
studying semiprimes composed of a pair of related primes, e.g. a pair of twin
primes. Artin's primitive root conjecture can be formulated in the new context.
Trigonometric polynomials based on chords and related to the new congruence
relation lead to new insights into the minimal polynomials of $2\cos(2\pi/n)$
and their relation to cyclotomic polynomials.
