Can bhavanga-consciousness be compared to the non-dual mind-state rigpa? While reading Pa Auk Sayadaw I came across the following:


  The bhavanga-consciousness is bright and luminous, and looks like a mirror in the heart: that is the mind-door.


This seems to have qualities similar to:


  In Dzogchen the fundamental, inherent nature of everything is called the "Ground Luminosity" or the "Mother Luminosity." 


Are these two qualities of mind related?
Generally speaking it is difficult to compare assertions made about the mind from different cultures many centuries apart. 

For example the Theravādins invented the bhavaṇgacitta for a specific purpose. There is a fundamental problem in Buddhist philosophy. Pratītyasamutpāda tells us that conditions must be present for effects to manifest (imassmim sati idam hoti) and that when the condition ceases the effect also ceases (imassa nirodhā idaṃ nirujjhati). However the idea of karma requires that effects (vipāka) manifest long after the condition (kamma) has ceased. The two do not work together. I call this the problem of Action at a Temporal Distance.

In order to get around this early Buddhists invented several different new ideas. The Theravādins opted for the the idea that each dhamma is very short-lived and acts as the condition for a similar dhamma to arise immediately afterwards. This creates an unbroken chain of short-lived mental events that accounts for karmic effects long after the action that set them in motion. 

But it has a number of downstream problems.

One is that we go to sleep everyday and consciousness stops in deep sleep. The bhavaṅgacitta is like a neutral gear for the mind, it keeps ticking over while the mind is inactive when we are asleep or at other times such as deep meditation when sense experience stops, or even when we stare vacantly at the view and there are no thoughts in our minds. 

Another problem the bhavaṅgacitta solves is that a wholesome mental event (kusala citta) cannot give rise to an unwholesome mental event (akusala citta). And because our minds alternate between kusala and akusala cittas there needs to be some way to get from one to the other. So the Theravādins say that when this happens a moment of bhavaṅgacitta has to be interposed. 

The fact is that while this was satisfactory to medieval Buddhists the scheme doesn't really work, even on it's own terms. See The Logic of Karma.

I'm much less familiar with the Dzogchen idea of "Ground Luminosity", but it looks to be influenced by another solution to the problem of Action at a Temporal Distance, the Yogācāra. In this view a metaphor from early Buddhism was taken up by Vasubandhu and reified by his followers. They liken an action to a seed and invoke the analogy of how the seed is planted in the ground and watered and eventually grows into a tree. Early Buddhists often used analogies from nature to try to explain their theories. The metaphor is that kamma is a seed. And of course the analogy requires a metaphorical "ground" for the seed to be planted in. And so the ālayavijñāna was invented as the metaphorical ground for the metaphorical seed to complete the analogy. Unfortunately Buddhists often lose sight of metaphors and reify them. The metaphor of ālayavijñāna as ground for karma seeds, became a real thing - an aspect of our minds that underlies all consciousness. Never mind that this becomes more or less identical to Brahmanical ideas of ātman/brahman and is thus eternalist. Not only does the underlying basis of consciousness become a real thing, but some meditative experiences are interpreted as contact with this luminous substrate - just as Brahmins had earlier mistaken such experiences for direct contact with Brahman centuries earlier. 

So no, the two ideas can't really be compared. Although at root they seem to stem from the problem of Action at a Temporal Distance and they both amount to reified metaphors. They may even amount to two different traditions interpreting similar meditative experiences according to their ideologies. But the interpretations themselves are not the same.