  Twisted hypercubes are generalizations of the Boolean hypercube, obtained by
iteratively connecting two instances of a graph by a uniformly random perfect
matching. Dudek et al. showed that when the two instances are independent,
these graphs have optimal diameter.
  We study twisted hypercubes in the setting where the instances can have
general dependence, and also in the particular case where they are identical.
We show that the resultant graph shares properties with random regular graphs,
including small diameter, large vertex expansion, a semicircle law for its
eigenvalues and no non-trivial automorphisms. However, in contrast to random
regular graphs, twisted hypercubes allow for short routing schemes.
