The general notion prevailing about Ardhanārīśvara, as mentioned in Puranas, is as follows:
Ardhanārīśvara is a composite androgynous (having both male and female characteristics) form of Shiva and his consort Parvati. This form is shown as a fusion of half-male and half-female forms, split down in the center.
===
Though Rig Veda contains, in general, an indirect references of अर्धनारीश्वर- Ardhanārīśvara concept here and there, as in Rig Veda 5.48 and Rig Veda 7.96.6, it is in Rig Veda 10.5.7 that contain the first direct reference to this concept.
Agni was stated as bull (Masculine) as well as a cow (Feminine).
असच्च सच्च परमे व्योमन्दक्षस्य जन्मन्नदितेरुपस्थे । अग्निर्ह नः प्रथमजा ऋतस्य पूर्व आयुनि वृषभश्च धेनुः ॥
asac ca sac ca parame vyoman dakṣasya janmann aditer upasthe | agnir ha naḥ prathamajā ṛtasya pūrva āyuni vṛṣabhaś ca dhenuḥ ||English translation:
Both the nonexistent and the existent are in the highest heaven, at the birth of the skillful one in the lap of Aditi. Agni is for us the first-born of truth, both a bull and a milk-cow in earlier times.
====
Śvetāśvataropaniṣad Chapter 4 shloka 10 mentions this concept of अर्धनारीश्वर - Ardhanārīśvara.
मायां तु प्रकृतिं विद्यान्मायिनं च महेश्वरम् ।
तस्यवयवभूतैस्तु व्याप्तं सर्वमिदं जगत् ॥ १०॥
Know then that Nature is Mayā, and that the great God is the Lord of Mayā. The whole world is filled with beings who form His parts.
