Earlier work took as universal mapping class group the collection \(\mathrm {PPSL}(2,{\mathbb Z})\) of all piecewise \(\mathrm {PSL}(2,{\mathbb Z})\) homeomorphisms of the unit circle \(S^1=\partial {\mathbb D}\) with finitely many breakpoints among the rational points in \(S^1\) . The spin mapping class group P(SL(2, \({\mathbb Z}\) )) introduced here consists of all piecewise-constant maps \(S^1\to \mathrm {SL}(2,{\mathbb Z})\) which projectivize to an element of \(\mathrm {PPSL}(2,{\mathbb Z})\) . We also introduce a spin universal Teichmüller space \(\mathcal {T}ess^+\) covering the earlier universal Teichmüller space \(\mathcal {T}ess\) of tesselations of \({\mathbb D}\) with fiber the space of \({\mathbb Z}\) /2 connections on the graph dual to the tesselation in \({\mathbb D}\) . There is a natural action which is universal for finite-type hyperbolic surfaces with spin structure in the same sense that is universal for finite-type hyperbolic surfaces. Three explicit elements of P(SL(2, \({\mathbb Z}\) )) are defined combinatorially via their actions on \(\mathcal {T}ess^+\) , and the main new result here is that they generate P(SL(2, \({\mathbb Z}\) )). Background, including material on hyperbolic and spin structures on finite-type surfaces, is sketched down to first principles in order to motivate the new constructions and to provide an overall survey. A companion chapter to this one gives a finite presentation of the universal spin mapping class group \(\mathrm {P}(\mathrm {SL}(2,{\mathbb Z}))\) introduced here.