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A compound of 4 piecewise linear (PL) M\"obius strips in$[0,3]^3\setminus[1,2]^3\subset\mathbb{R}^3$ and the Holden-Odom-Coxeterpolylink of 4 locked hollow equilateral triangles are seen equivalent from agroup-theoretical point of view, where each PL M\"obius strip has a minimum offacets, maximum area and boundary formed by orthogonal segments withend-vertices in $\mathbb{Z}^3$. This leads to 1-factorizations of the 4-cubegraph $Q_4$ and its antipodal quotient $K_{4,4}$, in the first case with each4-cycle having an edge in common with each 1-factor. Toroidal subgraphs of$Q_4$ and $K_{4,4}$ are characterized and an Euler square is associated.

Piecewise Linear M\"obius Strip, 4-Triangle Polylinks and 1-Factorizations of $Q_4$ and $K_{4,4}$