<p>A mathematical model is presented which creates four novel classes of <i>m</i>-fold cyclic twins in two dimensions, in addition to one class previously known. Each member of a class is characterized by four integer parameters: the twin modulus <i>m</i>, naturally describing the rotational symmetry of the twin, and three more parameters, <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(\mu \)</EquationSource> <EquationSource Format="MATHML"><math> <mi>μ</mi> </math></EquationSource> </InlineEquation>, <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\(\nu \)</EquationSource> <EquationSource Format="MATHML"><math> <mi>ν</mi> </math></EquationSource> </InlineEquation>, and <InlineEquation ID="IEq3"> <EquationSource Format="TEX">\(\sigma \)</EquationSource> <EquationSource Format="MATHML"><math> <mi>σ</mi> </math></EquationSource> </InlineEquation>, respectively, related to the properties of an integer inclination sequence fundamental for the generation of the cyclic twins by an <i>m</i>-fold composition of discrete circle involute spirals. A selection of the most interesting cases is depicted for each class, and a number of potential applications are discussed.</p>