"The difference of the two earrings is of course of utmost iconographic-iconologic importance:  the fact that one earring is the patrakuNDala,  i.e. the twisted golden palm leaf in the left ear-lobe as worn by village belles,  whereas the other earring is the makarakuNDala,  i.e. the man's ornamental earring in the shape of a crocodile,  this fact symbolizes the unity of the male and the female,  the dual nature of the Lord,  the ardhanArIs'vara concept."

In the footnotes to patrakuNDala,  he says "other name:  karNapatra,  Ta. Olia".

In the next footnote,  he says "Sometimes,  the right earlobe of  naTarAja remains free of ornament."

Harsha V. Dehejia wrote:

Nataraja wears a female earring in his left ear (kundala) while his right ear, though pierced, is bare. One can explain this by child like stories which say that his right earring fell off while dancing. The truth of the matter is that Nataraja is an ardhanirshvara and therefore must express his feminine aspect.  This is stylised by the kundala. Nataraja cannot be in a state of ananda (bliss) without a dynamic harmony of purusha and prakriti, Shiva and Parvati, male and female. This is the foundational doctrine of Kashmir Shaivism.