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We show that the 56-vertex Klein cubic graph $\G'$ can be obtained from the28-vertex Coxeter cubic graph $\G$ by 'zipping' adequately the squares of the24 7-cycles of $\G$ endowed with an orientation obtained by considering $\G$ asa $\mathcal C$-ultrahomogeneous digraph, where $\mathcal C$ is the collectionformed by both the oriented 7-cycles $\vec{C}_7$ and the 2-arcs $\vec{P}_3$that tightly fasten those $\vec{C}_7$ in $\G$. In the process, it is seen that$\G'$ is a ${\mathcal C}'$-ultrahomogeneous (undirected) graph, where${\mathcal C}'$ is the collection formed by both the 7-cycles $C_7$ and the1-paths $P_2$ that tightly fasten those $C_7$ in $\G'$. This yields anembedding of $\G'$ into a 3-torus $T_3$ which forms the Klein map of Coxeternotation $(7,3)_8$. The dual graph of $\G'$ in $T_3$ is the distance-regularKlein quartic graph, with corresponding dual map of Coxeter notation $(3,7)_8$.

From the Coxeter graph to the Klein graph