The "official" Grand Antiprism is a uniform polychoron consisting of 20 pentagonal antiprisms in two orthogonal rings, with the rings joined to each other via tetrahedra. These two rings are just like the mutually orthogonal rings in the duoprisms and the duocylinder, and the tetrahedra are laid out along a toroidal 2-manifold just like the duocylinder's ridge.
These likenesses are not coincidental. If we relax the regularity requirement on the tetrahedra, we may obtain a whole series of grand antiprisms, consisting of 2n m-gonal antiprisms in two rings of n antiprisms each, with non-regular tetrahedra connecting them. I'm going to denote these grand antiprisms as n,m-grand antiprisms, where n denotes the number of antiprisms (n must always be even, otherwise the shape is not closed by the sheet of tetrahedra), and m denotes the base polygon of the antiprisms.
Well, as n and m approach infinity, the n,m-grand antiprisms approach a shape with a similar geometry to the duocylinder, except that where the duocylinder's ridge is, this shape has another 3-manifold (basically a thickened version of the duocylinder's ridge), joined to the two tori by two ridges of the kind of shape as the duocylinder's ridge. The thickness of this 3-manifold depends on how much we "stretch" the tetrahedra in the approximating grand antiprisms.
We may think of this shape as the cantellated duocylinder: it's what we get if we truncate the duocylinder along its toroidal ridge. The two bounding tori shrink (but remains topologically the same), and a new 3-manifold is introduced. Or, alternatively, it's what we get if we expand the duocylinder's bounding tori outwards, and fill in the gap with a new 3-manifold (hence the name cantellated). The cantellated duocylinder is the limiting shape of the (non-uniform) grand antiprisms, and thus can be usefully approximated by an n,m-grand antiprism of a high order. (This is for the same reasons we use high-order duoprisms to approximate a duocylinder: so that we don't have to deal with quadratic 4D equations, which are Not Nice(tm) to program.)
The cantellated duocylinder is also the limiting shape of another related series of polychora: the cantellated duoprisms. Basically, you take a duoprism and insert cubes (well, cuboids) where its square faces are. The resulting shape as the same number of prisms as the duoprism, but with additional cuboidal cells. Taking the limit of these shapes as n and m grow to infinity in the source n,m-duoprism, we approach the cantellated duocylinder.