There exist myriads of off-shell worldline supermultiplets for $(N \leq 32)$- extended supersymmetry in which every supercharge maps a component field to precisely one other component field or its derivative. A subset of these extends to off-shell worldsheet $(p, q)$-supersymmetry, and is characterized herein by evading an obstruction specified visually and computationally by the "bow-tie" and "spin sum rule" twin theorems. The evasion of this obstruction is proven to be both necessary and sufficient for a worldline supermultiplet to extend to worldsheet supersymmetry; it is also a necessary filter for dimensional extension to higher-dimensional spacetime. We show explicitly how to "re-engineer" an Adinkra—if permitted by the twin theorems—so as to depict an off-shell supermultiplet of worldsheet $(p, q)$-supersymmetry. This entails starting from an Adinkra depicting a specific type of supermultiplet of worldline $(p+q)$- supersymmetry, judiciously re-defining a subset of component fields and partitioning the worldline $(p, q)$-supersymmetry action into a proper worldsheet $(p, q)$-supersymmetry action.
"On dimensional extension of supersymmetry: from worldlines to worldsheets." Adv. Theor. Math. Phys. 16 (6) 1619 - 1667, December 2012.