The Samayācāra
Samayācāra refers both to the internal mode of worship, and to worship employing hōmās, fire rituals. We can perform these homās and internal visualizations whether or not external puja articles are present. If they are not present, we can simply visualize them and still perform the entire pūjā. The Samayācāra traditions we follow come to us from the divya paramparā (divine lineage)—that is, through Bālājī: Bālā Tripurasundarī herself, who is our guru.
The Dakṣiṇācāra
In the Dakṣiṇācāra tradition, we worship the Śrī Cakra. It is a bāhya pūjā, meaning that you are worshiping something outside yourself; usually a vigraham, i.e., an idol or yantra. However, the suvāsinī puja is also done, the suvāsinī being a woman who represents the Śakti, though in this tradition the pūjā is done only to her feet.
I have been given the Mēdha Dakṣiṇāmūrti from the Siddha Paramparā, from Sarasvatī. So the Dakṣiṇācāra sampradāya has been given to us through the Sarasvatī Order. I am eligible for that, as are those who have taken dīkṣā from me.
The Kaulācāra
In the Kaulācāra tradition, the idol or the yantra is, in fact, a living woman or man, or a couple, or a circle of devotees. You can also think of her as the union of Śiva and Śakti. Because she is the life in all, you can worship her as a woman, as a man, or as both. There are no restrictions in this regard. When we give her a bath, we not only chant the Durgā and Lakṣmī Sūktams; we also chant the Puruṣa Sūktam and the Rudram.
Kaulas worship both Śiva and Śakti. It is said, regarding her mantra of 15 letters, that on the five days when "ka"s and "ha"s occur, one should worship a man; on the three days when hrīm comes, one should worship a couple; and on the seven remaining days, one should worship a woman.
I was given Samayācāra dīkṣā by Bālājī, and I was given the Dakṣiṇāmūrti tradition by Sarasvatī. The Kaulācāra traditions have also come to us from the Siddha Yoga paramparā. Kaulācāra traditions are from the Dattātrēya sampradāya, as codified in the Paraśurāma Kalpasūtra. In the Kaulācāra tradition, the notion of the self is completely absent. You see everyone as yourself. You invoke the Goddess into your wife, into the suvāsinī or into a mandala. You yourself become the Goddess in the Virajā Hōma, even as you are worshiping her.
Source: Gifts from the Goddess: Selected Works of Sri Amritananda Natha Saraswati by Michael M Bowden