Kyle Dixon, reddit:

This topic can get somewhat detailed but I think an important distinction to make is between what Nagarjuna called dependent existence [parabhava] and dependent origination [pratityasamutpada]. Nowadays, these two principles are often conflated, but if we consult Nagarjuna's writings on this topic, we find that he makes a firm distinction and that distinction is important.

Parabhava, or "dependent existence" as it is sometimes glossed, describes precisely this idea of things depending upon each other and arising in mutual dependence. Nagarjuna actually criticizes this idea and says that this view of existence is merely a guise for svabhava or inherent existence. By virtue of parabhava, the principle of svabhava sort of covertly sneaks into the fold and if it is not recognized, the individual may simply replace the misconception of svabhava with a view of parabhava.

Dependent origination [pratityasamutpada] is not actually things arising in mutual dependence, not necessarily. In dependent origination proper the idea of origination or arising should ideally but understood as being cradled in what these teachings call avidya, or ignorance. In Nagarjuna's Yukisastikakarika he states:

        When the perfect gnosis sees that things come from ignorance as condition, nothing will be objectified, either in terms of arising or destruction.

Going on to state:

        Since the Buddhas have stated that the world is conditioned by ignorance, why is it not reasonable [to assert] that this world is [a result of] conceptualization? Since it (the world) comes to an end when ignorance ceases; why does it not become clear that it was conjured by ignorance?

As such, phenomena appear to originate due to the presence of ignorance influencing the mindstream, polluting the mindstream, so that things are not seen accurately. Once ignorance is removed from the mindstream, then phenomena are seen to be primordially unoriginated, or non-arisen.

In this way the real meaning of "dependent origination" is that phenomena appear to originate in dependence upon the presence of ignorance. Apparent entities are dependently originated with ignorance. However in actuality there has never really been origination at any point in time, only the misconception of origination.

This correlation is made explicit in quite a few places. Manjusri states:

        Whatever is dependently originated does not truly arise.

Nagarjuna:

        What originates dependently is unoriginated!

Candrakirti:

        The perfectly enlightened buddhas proclaimed, "What is dependently created is uncreated [non-arisen]."

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John Tan commented:

 

We project svabhava onto what appears where it actually lacks.  Dependent existence is based on that cognitive fault and faulty premise whereas dependent orignation is taught to correct that.  It explains the process of formation upon ignorance as well as de-construction.  It is also the king of reasoning, seeing dependent origination also sees the 8 negations; therefore it is an enlightened view.

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