The incidence matrix of the real encasing polytope, the cover of 2 completely orthogonal n-gons (the dual of n,n-dip – esp. n = 4 being hex), is

 2n |  2  n |  3n | 2n
----+-------+-----+---
  2 | 2n  * |   n |  n
  2 |  * nn |   4 |  4
----+-------+-----+---
  3 |  1  2 | 2nn |  2
----+-------+-----+---
♦ 4 |  2  4 |   4 | nn

But here the inter-linking edges are to be considered as digons, that is, do not count as (real) 1-dimensional, but as complex 1-dimensional elements, and are the ones to be considered here. Further the linking triangles too are to be blown-up, becoming kind of a tet, but with one of its edges being reduced to zero length. – Therefore the incidence matrix rather runs like this, contradicting heavily to Euler's rule (for instance still having 4 such blown-up triangular "faces" incident to those blown-up "edges", instead, as normal, having only 2 cells per face):

 2n |  2  2n |  6n  n | 2n  3n
----+--------+--------+-------
  2 | 2n   * |  2n  0 |  n   n
  2 |  * 2nn |   4  1 |  2   4
----+--------+--------+-------
  3 |  1   2 | 4nn  * |  1   1
  2 |  0   2 |   * nn |  0   4
----+--------+--------+-------
♦ 4 |  2   4 |   4  0 | nn   *
  3 |  1   4 |   2  2 |  * 2nn

The incidence matrix of the complex polytope thus is

2n |  n
---+---
 2 | nn

Generators

with e  = exp(2πi/n),   E  = exp(2πi k/n),
where 1 < k < n and k not divisor of n

R0 = / 0 1 \ ,  R1 = / e 0 \
     \ 1 0 /         \ 0 E /

R02  =  1                                  (exchange)
R1n  =  1                                  (rotation-rotation*)
R0 * R1 * R0 * R1  =  R1 * R0 * R1 * R0    (up → rot.* → down → rot.  =  rot. → up → rot.* → down)