Recapitulation theory encapsulates a rather simple truth - that development often proceeds by strapping on extra developmental phases. When this happens, ontogeny does indeed recapitulate phylogeny.
The brain makes a simple example. Brains are divided into layers, and over evolutionary time, more layers have been added.The human neocortex, for example consists of six layers. During development the neural layers are formed progressively. Migrating neurons climb up a scaffolding made of radial glial cells and bypass previous layers of neurons in the cerebral cortex, creating a new layer on top of their predecessors. This mirrors development over evolutionary time - where ancestors had fewer layers of neurons.
A simple, pure example of cultural recapitulation theory can be found in knot theory. Knots are often formed by tying knows on top of other knots. The simpler knots come first historically as well as during knot construction. A reef knot is a simple example of one knot being tied on top of another knot in order to strengthen it.
The concept of "cultural recapitulation" is often used in another way. Some say that the way that individual learning mirrors historical discovery is a case of cultural recapitulation. For example, the order in which scientific concepts are taught to children might mirror the order in which they are discovered. This is certainly recapitulation and it does involve culture, but the analogy with Ernst Haeckel's idea is weaker than with the knot example I gave. The reason is that the development involved is the development of a young animal - a memetic host rather than a memetic product. The corresponding biological analogy would be if the way in which organisms were infected with pathogens reflect the sequence in which those pathogens evolved. If is easy to imagine reasons why that might be true - for example new pathogens might be less expert at spreading to new hosts. However, this seems a bit different to classical recapitulation theory - since that does not normally involve symbiosis.
Will cultural recapitulation theory suffer the same fate as its organic counterpart? It is, perhaps too early to tell. However, maybe cultural recapitulation theory will help to revitalize recapitulation theory in the organic domain. It is a bit of a shame that recapitulation theory is so widely labelled as a dud idea. Perhaps cultural examples will help to illuminate the truth at the core of the idea.