Thanks for your question. I specialize in Twin Flames and soulmates and have been in union with my own Twin Flame for a long time so I'll try to help.
Kundalini awakening, also known as kundalini rising, is simply the awakening of the scared energy and is not exclusive to twin flames, and occurs in all soul beings as it pertains to their own ascension process. It occurs at different times for different soul beings.
Its important due to the energy exchange between twins, because twins share a chakra system and because of the awakening and ascension process which the twins embark on simultaneously and in tandem. Additionally, the twins telepathy and other gifts awaken and increase during these processes of ascension and awakening.
Kundalini awakening in Twin Flames refers to the shared chakra system versus being connected through chakra cords as typically soulmates connect to one another. While the twins do share a chakra system, twins still serve as soulmates for others and are able to connect chakra cords to others as well. Because the twins share a chakra system, some find themselves inadvertently connected through chakra cords to people they didn't intend to connect with due to the connection of their twin flame partner or counterpart.
The Kundalini awakening also referred to as Kundalini rising is not exclusive to twin flames and happens for everyone during the Ascension process. The Kundalini awakening is simply the awakening of sacred energy at the sacral chakra. It's an aspect of soul evolution during ascension. Because twin flames share a chakra system although twins are a divided soul housed in 2 separate human bodies, the Kundalini awakening typically happens simultaneously in Twin Flames.
Twin Flames are two halves of the same soul incarnate in two separate human bodies. During this, the 3rd and final incarnation the soul is split into the two halves: one male energy and the other half female energy. Twin Flames are complimentary; one possessing all of the ABC traits while his partner will possess all of the XYZ traits. Like 2 perfect mirrored compliments. 2 pieces to a puzzle that only contains 2 pieces.
The energy exchange between twins is important to reference. As this awakening transpires, this can cause the illumination of a previously UNAWAKENED twin flame to awaken, again, due to the process of ascension and during the process of Kundalini awakening or rising. During this process, soul beings find that they discover certain abilities they did not previously realize they have, and other gifts are developed.
I suggest researching the symptoms of Ascension and chakra connections to better understand how these processes work and what to expect. There are some very helpful and insightful articles that are free to read in the blog section of my site which I've made available to the public with the objective to assist others through their journey and to achieve twin flame union more easily. Please feel free to take a look.
The Kundalini awakening occurs at the sacral chakra — again, twin flames share a chakra system — so Tantra comes easily for the twins including distance Tantra. The telepathic connection that occurs by default between the twins also increases at this time of awakening.
I hope this helps!