Each of the ten mudrAs starting with sarvasaMkShobhiNI and ending with sarvatrikhaNDA have their own importance. However, trikhaNDA is indeed special due to one reason. The entire kalApa of bahiryAga assumes significance after the presence of shrI sundarI in the bimba or yantra and the mudrA which accomplishes the AvAhana of parAmbA is trikhaNDA:
trikhaNDeyaM mahAmudrA tripurAhvAnakarmaNi [vAmakeshvare]
Hence, bahiryAga siddhi is attributed to trikhaNDA mudrA specifically among the dasha mudrAs. Of course, bhAskararAya does state in setubandha that upAsanA is japarUpa in kali yuga and hence archana is an anga of japa and not vice versa. If this is accepted, then bahiryAga siddhi would seem more logical when associated with sarvayoni mudrA rather than trikhaNDA. However, this view, even if not incorrect, does not seem to find support either in kalpasUtra, tantrarAja or nityAShoDashikArNava where japa is an anga of bahiryAga. So also says the tradition.
Here, tri means three and it can denote either janma, mR^ityu and jarA or sattva, rajas and tamas. As this mudrA, which is non-different from parAmbA, destroys this triad, it is called trikhaNDA:
trIn khaNDayatIti trikhaNDA ||
In other words, the mudrA grants kaivalya or mokSha.
Alternately, tri here represents icchA, j~nAna and kriyA shaktis or the three khaNDas of the supreme mahAvidyA namely vAgbhava, kAmarAja and soma kUTas. The mudrA which represents these three khaNDas is named trikhaNDA.
mudrAkhyA sA yadA samvidambikA trikalAmayI |
trikhaNDArUpamApannA sadA sAnnidhyakAriNI || [yoginI tantre]
Though vAmakeshvara tantra teaches only two khaNDas for the mudrA, bhAskararAya instructs the upAsaka to refer to tantrAntara and necessarily accomplish khaNDatraya:
trikhaNDeti yogArthasampattaye tantrAntaraM sharaNIkR^itya khaNDatrayamavashyaM sampAdanIyam |
It is of great significance to know why a specific mantra of three kUTas is coupled with trikhaNDA during AvAhana of parAmbA and not panchadashI or ShoDashI. The answer to this lies in the concept of awakening kuNDalinI or the serpent of fire which is what AvAhana is all about. And when we say snake, there is hissing and she is yogAgni svarUpinI. I am sure many have seen the Harry Potter series of movies. Potter, to communicate with a snake (in Goblet of Fire), uses its language which is predominantly a hissing sound. While this is fiction, the Yogis of Tibet who practice gtummo to awaken the sacred fire use a similar technique. The exact sanketa rahasya should be grasped from Sadguru.
Also, one can refer to the following statement from bhAvanopaniShad:
AdhAranavakaM mudrAshaktayaH ||
The nine AdhAras are mUlAdhArAdi ShaTka, Urdhva and adhaH sahasrAra, and lambikAgra. Each of these corresponds to one of the nine mudrAs (sarvasaMkShobhiNI to sarvayoni). The samaShTi of these navAdhAras and hence of nava mudrAs (grouped into three categories) is represented by trikhaNDA mudrA. Thus, the mudrA also represents the source tattva of the nine aspects: sR^iShTi sR^iShTi, sR^iShTi sthiti etc. shrI bhAskararAya thus illustrates the following prayoga:
pAdA~NguShTharUpa AdhAranavakAtmane sarvatrikhaNDAmudrAyai namaH ||
mukundo govindo janakatanayAlAlitapadaH
padaM prAptA yasyAdhamakulabhavA chApi shabarI |
ramAnAtho rAmo ramatu mama chitte tu satatam ||