Ardhanarīśvara
An important aspect of Naṭarāja, and for that matter of Śiva Himself, is His being Ardha-nārīśvara ("The Lord Who Is Half Woman"). According to this symbolism, the Unitary Principle divides itself vertically: the right-hand side is masculine/Male, and the left-hand side feminine/Female. We find corroborations for this from various texts, and even in Manu-smṛti.
Manusmṛti 1.32:
dvidhā kṛtvā''tmano deham ardhena puruṣo'bhavat | ardhena nārī ||
द्विधा कृत्वाऽऽत्मनो देहम् अर्धेन पुरुषोऽभवत् । अर्धेन नारी ॥
Also, Bṛhadāraṇyakopaniṣad 1.4.3: ātmānaṁ dvedhā'pātayat;
and even in a play of Bhāsa. See Avimāraka 2.12:
pūrvā tu kāṣṭhā timirā'nuliptā sandhyā'ruṇā bhāti ca paścimā''śā |
dvidhā vibhaktā'ntaram antarikṣaṁ yāty ardha-nārīśvara-rūpa-śobhām ||
पूर्वा तु काष्ठा तिमिराऽनुलिप्ता सन्ध्याऽरुणा भाति च पश्चिमाऽऽशा ।
द्विधा विभक्ताऽन्तरम् अन्तरिक्षं यात्यर्ध-नारीश्वर-रूप-शोभाम् ॥
The idea occurs in Kālidāsa's Kumārasambhava with specific reference to Śiva.
Kumārasambhava 2.7:
strī-puṁsāv ātma-bhāgau te bhinna-mūrtes sisṛkṣayā;
स्त्री-पुंसाव् आत्म-भागौ ते भिन्न-मूर्तेस् सिसृक्षया;
Cf. Mālavikāgnimitra 1.4:
Rudreṇedam Umākṛta-vyatikare svāṅge vibhaktaṁ dvidhā; रुद्रेणेदम् उमाकृत-व्यतिकरे स्वाङ्गे विभक्तं द्विधा;
also 1.1: kāntā-saṁmiśra-deho'py aviṣaya-manasāṁ yaḥ parastād yatīnām. कान्ता-संमिश्र-देहोऽप्यविषय-मनसां यः परस्ताद् यतीनाम्।
In the image under our consideration, Naṭarāja exhibits a conspicuous orientation of the limbs towards the right side, as all except one (the hind left hand), are turned that way. The bent left leg is also stretched towards the right. Once again, this orientation towards the right emphasizes Naṭarāja's dominant Puruṣa aspect.
Naṭarāja's ears present a study in contrast, between the masculine and the feminine. The left ear has a patra-kuṇḍala ("Leaf-Ornament"), worn typically by women, and the right ear has a makara-kuṇḍala (Crocodile Ornament), worn typically by men, even though the latter is not so easily noticeable. A double-string necklace adorns His throat, easily discernible as being the maṅgala-sūtra, the "Auspicious String" of a woman, which also goes to emphasize once again the Ardha-nārīśvara aspect. This masculine-feminine contrast is also evident in the depiction of the fingers of the left hands and the right hands, as too, the toes of the left foot and the right foot: the ones on the left look slenderer, tenderer, and more graceful.