The transfer or share of knowledge between languages is a popular solution to resource scarcity in NLP. However, the effectiveness of cross-lingual transfer can be challenged by variation in syntactic structures. Frameworks such as Universal Dependencies (UD) are designed to be cross-lingually consistent, but even in carefully designed resources trees representing equivalent sentences may not always overlap. In this paper, we measure cross-lingual syntactic variation, or anisomorphism, in the UD treebank collection, considering both morphological and structural properties. We show that reducing the level of anisomorphism yields consistent gains in cross-lingual transfer tasks. We introduce a source language selection procedure that facilitates effective cross-lingual parser transfer, and propose a typologically driven method for syntactic tree processing which reduces anisomorphism. Our results show the effectiveness of this method for both machine translation and cross-lingual sentence similarity, demonstrating the importance of syntactic structure compatibility for boosting cross-lingual transfer in NLP.