Since, in every group each element has an associated inverse, it follows that every group contains an antitonic "spine".

Consider, as an example, the dihedral group, D₃ (symmetries of equilateral triangle), in another file at this Website:

CAYLEY TABLE OF D₃

o	I	R1	R2	M1	M2	M3
I	I	R1	R2	M1	M2	M3
R1	R1	R2	I	M2	M3	M1
R2	R2	I	R1	M3	M1	M2
M1	M1	M3	M2	I	R2	R1
M2	M2	M1	M3	R1	I	R2
M3	M3	M2	M1	R2	R1	I

In this Table, please note where row and column intersect in the identity element, I. These intersections constitute the antitonic "spine" of D₃:

	I I = I
	R1R2 = I
	R2R1 = I
	M1M1 = I
	M2M2 = I
	M3M3 = I

Although tedious to do so, this could be demonstrated for any continuous group.