Namastripurabhairavyai ||
Each of the ten mudrās starting with Sarvasaṅkṣōbhiṇī and ending with Sarvatrikhaṇḍā has its own importance. However, Trikhaṇḍā is of special importance. The entire kalāpa of Bahiryāga assumes significance after the presence of Mahātripurasundarī in the Bimba or yantra and the mudrā which accomplishes the āvāhana of Parāmbā is Trikhaṇḍā:
trikhaṇḍēyaṁ mahāmudrā tripurāhvānakarmaṇi || (Vāmakēśvara Tantra)
Hence, Bahiryāga siddhi is attributed to Trikhaṇḍā mudrā specifically among the daśa mudrās. Of course, Bhāskararāya does state in Sētubandha that upāsanā is primarily japarūpa in Kaliyuga and hence archana is an aṅga of Japa and not vice versa. If this is accepted, then Bahiryāga siddhi would seem more logical when associated with Sarvayōni mudrā rather than Trikhaṇḍā as Yōnimudrā removes the dōṣas during Japa. However, this view, even if not incorrect, does not seem to find support either in Kalpasūtra, Tantrarāja or Nityāṣōḍaśikārṇava where Japa is presented as an aṅga of Bahiryāga.
Here, ‘tri’ means three and it can denote either janma, mr̥tyu, and jarā or sattva, rajas, and tamas. As this mudrā, which is non-different from the blissful aspect of Parāmbā, destroys this triad, it is called Trikhaṇḍā:
trīn khaṇḍayatīti trikhaṇḍā ||
In other words, the mudrā grants Mōkṣa in its highest expression.
Alternately, ‘tri’ here represents iccā, jñāna, and kriyā śaktis or the three khaṇḍas of the Mahāvidyā namely Vāgbhava, Kāmarāja and Sōma kūṭas. The mudrā which represents these three khaṇḍas is named Trikhaṇḍā.
mudrākhyā sā yadā samvidambikā trikalāmayī |
trikhaṇḍārūpamāpannā sadā sānnidhyakāriṇī || (Yōginī Tantra)
Though the Vāmakēśvara Tantra teaches only two khaṇḍas for the mudrā, Bhāskararāya instructs the upāsaka to refer to tantrāntara and necessarily accomplish khaṇḍatraya:
trikhaṇḍēti yōgārthasampattayē tantrāntaraṁ śaraṇīkr̥tya khaṇḍatrayamavaśyaṁ sampādanīyam ||
It is of great significance to know why a specific mantra of three kūṭas is coupled with Trikhaṇḍā during the āvāhana of Parāmbā and not Pañcadaśī or ṣōḍaśī. The answer to this lies in the concept of awakening Kuṇḍalinī or the serpent of fire which is what āvāhana is all about. And when we say a snake, there is hissing and she is Yōgāgnisvarūpinī. The Yogis of Tibet who practice gTummo to awaken the sacred fire use a similar concept, albeit a different technique. The exact saṅkēta rahasya here should be grasped from Sadguru.
Also, one can refer to the following statement from Bhāvanōpaniṣad:
ādhāranavakaṁ mudrāśaktayaḥ ||
The nine ādhāras are Mūlādhārādi ṣaṭka, ūrdhva and adhaḥ Sahasrāra, and Lambikāgra. Each of these corresponds to one of the nine mudrās (Sarvasaṅkṣōbhiṇī to Sarvayōni). The samaṣṭi of these navādhāras and hence of the navamudrās (grouped into three categories) is represented by Trikhaṇḍā mudrā. Thus, the mudrā also represents the source tattva of the nine aspects: sr̥ṣṭi-sr̥ṣṭi, sr̥ṣṭi-sthiti, etc. Bhāskararāya thus illustrates the following prayōga:
pādāṅguṣṭharūpa ādhāranavakātmanē sarvatrikhaṇḍāmudrāyai namaḥ ||