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We revisit the the large spin asymptotics of 15j symbols in terms of cosinesof the 4d Euclidean Regge action, as derived by Barrett and collaborators usinga saddle point approximation. We bring it closer to the perspective ofarea-angle Regge calculus and twisted geometries, and compute explicitly theHessian and phase offsets. We then extend it to more general SU(2) graphinvariants, showing that saddle points still exist and have a similarstructure. For graphs dual to 4d polytopes we find again two distinct saddlepoints leading to a cosine asymptotic formula, however a conformalshape-mismatch is allowed by these configurations, and the asymptotic action isthus a generalisation of the Regge action. The allowed mismatch correspond toangle-matched twisted geometries, 3d polyhedral tessellations with adjacentfaces matching areas and 2d angles, but not their diagonals. We study thesegeometries, identify the relevant subsets corresponding to 3d Regge data andflat polytope data, and discuss the corresponding Regge actions emerging in theasymptotics. Finally, we also provide the first numerical confirmation of thelarge spin asymptotics of the 15j symbol. We show that the agreement isaccurate to the per cent level already at spins of order 10, and thenext-to-leading order oscillates with the same frequency and same global phase.

SU(2) graph invariants, Regge actions and polytopes