In the context of Linked Open Data, substantial progress has been made in mining of property subsumption and equivalence axioms. However, little progress has been made in determining if a predicate is symmetric or if its inverse exists within the data. Our study of popular linked datasets such as DBpedia, YAGO and their associated ontologies has shown that they contain very few inverse and symmetric property axioms. The state-of-the-art approach ignores the open-world nature of linked data and involves a time-consuming step of preparing the input for the rule-miner. To overcome these shortcomings, we propose a schema-agnostic unsupervised method to discover inverse and symmetric axioms from linked datasets. For mining inverse property axioms, we find that other than support and confidence scores, a new factor called predicate-preference factor (ppf) is useful and setting an appropriate threshold on ppf helps in mining quality axioms. We also introduce a novel mechanism, which also takes into account the semantic-similarity of predicates to rank-order candidate axioms. Using experimental evaluation, we show that our method discovers potential axioms with good accuracy. © 2017, Springer International Publishing AG.