WordNet records no synonym and no antonym for sphinx. It is the only member of each of its synsets, which is common for technical, taxonomic and single-sense vocabulary — there is no other English word that means the same thing. The broader and narrower terms below are the useful relations in that case.
Counts are distinct word forms across every sense. A synonym that appears under three senses is counted once here and shown three times below, under each sense it belongs to.
an inscrutable person who keeps his thoughts and intentions secret
(Greek mythology) a riddling winged monster with a woman's head and breast on a lion's body; daughter of Typhon
one of a number of large stone statues with the body of a lion and the head of a man that were built by the ancient Egyptians
Synonyms, antonyms and the broader and narrower relations for sphinx come from Princeton University’s WordNet. Full sources and licences.