  This paper is part of a program initiated by Saharon Shelah to extend the
model theory of first order logic to the non-elementary setting of abstract
elementary classes (AECs). An abstract elementary class is a semantic
generalization of the class of models of a complete first order theory with the
elementary substructure relation. We examine the symmetry property of splitting
(previously isolated by the first author) in AECs with amalgamation that
satisfy a local definition of superstability.
  The key results are a downward transfer of symmetry and a deduction of
symmetry from failure of the order property. These results are then used to
prove several structural properties in categorical AECs, improving classical
results of Shelah who focused on the special case of categoricity in a
successor cardinal.
  We also study the interaction of symmetry with tameness, a locality property
for Galois (orbital) types. We show that superstability and tameness together
imply symmetry. This sharpens previous work of Boney and the second author.
