Learning the structure of a Bayesian network (BN) from data is NP-hard. To efficiently handle high-dimensional datasets, many BN local structure learning algorithms are proposed. These learning algorithms can be categorized into two types: constraint-based and score-based. These learning algorithms learn the local structures separately for each node. As a result, asymmetric pairs of neighbors and Markov blankets create conflicts between the local structures. To resolve the conflicts, symmetry correction is required. The commonly used AND-rule symmetry correction method, which simply drops nodes in asymmetric pairs from the neighbor sets and Markov blankets of both nodes, may result in loss of information in learning the BN. In this paper, we propose a hybrid framework that combines a local structure learning algorithm of a particular type (either constraint-based or score-based) with a data-driven symmetry correction method of the other type. The score-based symG method and the constraint-based symC method are proposed to be used in the hybrid framework. Empirical results show that performances of constraint-based learning algorithms are improved by using the proposed score-based symG method. Similarly, the performance of score-based local learning algorithm is better when symC is used, compared to using symG.