The hyponymy relation (and converse hypernymy) which forms the ISA backbone of taxonomies and ontologies such as WordNet (Fellbaum, 1989), and determines lexical entailment (Geffet and Dagan, 2005), is asymmetric.
the cohyponymy relation which relates two words unrelated by hyponymy but sharing a (close) hypernym, is symmetric, as are synonymy and antonymy.