It is given a Schläfli symbol h{4,3...3,4} representing the regular form with half the vertices removed and containing the symmetry of Coxeter group
can be created by removing another mirror on an order-4 peak.
[1] The alternated hypercube facets become demihypercubes, and the deleted vertices create new orthoplex facets.
The vertex figure for honeycombs of this family are rectified orthoplexes.
These are also named as hδn for an (n-1)-dimensional honeycomb.