Kayne (1994) and Moro (2000) argue that an asymmetric c-command relation between syntactic constituents X and Y maps onto linear precedence relations between the lexical items in X and Y respectively, the Linear Correspondence Axiom (LCA). Kayne and Moro demonstrate that this assumption, together with the properties of linear orderings (totality, asymmetry and transitivity), imposes linguistically significant limits on the class of possible syntactic structures. In this short paper, we explore an alternative approach to deriving restrictions on syntactic structure that depends not on a specific assumption about linearization, but instead focuses directly on the non-existence of structural symmetry. Specifically, we consider the class of trees, NoAut, which admit no (non-trivial) automorphism (nonidentity mappings between the nodes that preserve structural relations). We demonstrate that NoAut imposes restrictions similar to the ones derived from the LCA, which have proven useful in syntactic theory. However, we prove that the LCA class is a proper subset of NoAut class. We briefly consider the linguistic relevance of the extra flexibility of the NoAut class.