I think that the twelve links of DO is a highly formalised presentation of Lord Buddhas teachings on anatta. In my view, the essence is that there is this body, and everything that is experienced is to be viewed as originating from this body. DO, the five khandhas, the four satipatthanas, all of these teachings is about this view.
Before the Buddhas time, everyone with a spiritual view of life believed that there is something behind this changing body that is the True Self. The Buddha summarised these views in the five khandhas - body, perceptions, feelings, sankharas and conciousness. So one view could be that a highly pleasent feeling was the True Self. The Buddha, however, said that all of the khandhas are to be viewed as originating from this body, and that this is the end of dukkha.
Look at this body and see that it is impermanent. You know that it is going to die and cease to exist as this body. Based on this insight you view the body as something alien, it doesn’t belong to your true nature. Right now there is a human body in the middle of your experience, and another time it can be an animal. So, you push away the body-nature - “body-nature isn’t me”. Now the other sects look for what IS the self - can it be feelings, conciousness or something else? But the Buddha said that whatever is experienced, that is to be viewed as originating from this body. And because the body is alien, everything that originates from the body is to be seen as alien too. It is like a process where you first push the body out of the “I”, or “me”, you view it as alien to your self. Then you push everything else out from the “I”, or “me”, based on the view that everything is connected with the body. When there is nothing left for the “I” or “me” to be based on, the only thing that remains is emptiness.
So, you no longer identify with the khandhas, but because you don’t identify with something new that you have found behind the changing things in the world, that is apart from the khandas, you don’t say that you are something that is apart from the khandhas. You are neither the khandas, nor anything outside the khandas, nor anything between. In the Buddhas words, “when there is no you there, you are neither here nor yonder nor between the two. This, just this, is the end of stress.” (SN 35.95)
In some suttas the Buddha says that there is the sense organs, the sense objects, the sense conciousness and the contact between these three. When there is an ignorance-element in this contact then there is sankharas too, and so on. You see that this is DO, but not explained as a sequence where one factor follows another in line. This 12 factors explanation is not as nature works, it is a list, and a list doesn’t explain the complexity of nature. This is just a way of formalising the teaching in a form that is easy to remember. The factors of DO happens all of them right now, but not in a twelve link sequence. You experience the world with ignorance and based on that there is craving connected with the body which you ignorantly view as “I” or “me”, and based on that craving there is performence of actions. When there is actions there is future birth, suffering and so on.