The Magic Encyclopedia Dudeney Patterns (note: transposed versions are also found in order 4 squares) 


pattern  horizontal  vertical 
I  2,3,0,1  2,3,0,1 
II  1,0,3,2  1,0,3,2 
III  3,2,1,0  3,2,1,0 
IV  0,1,2,3  1,0,3,2 
V  0,1,2,3  2,3,0,1 
VI  0,1,2,3  3,2,1,0 
VII  0,1,2,3  {(3,2,1,0),(1,0,3,2),(1,0,3,2),(3,2,1,0)} 
VIII  0,1,2,3  {(2,3,0,1),(3,2,1,0),(3,2,1,0),(2,3,1,0)} 
IX  0,1,2,3  {(1,0,3,2),(3,2,1,0),(3,2,1,0),(1,0,3,2)} 
X  0,1,2,3  {(3,2,1,0),(2,3,0,1),(2,3,0,1),(3,2,1,0)} 
XI  1,0,3,2  0,3,2,1 
XII  2,3,0,1  0,1,3,2 
Theoretical basis  
The use of "self inverse permutations" to desribe complementairy pairs within a square gives us some analitic handles to describe the pair. The combination of a horizontal and vertical permutation describe a pair of cells where complementairy pairs reside provided that the horizontal (H) and vertical (V) obey the simple rule H[i] <> V[i] (otherwise the combination would map a cell onto itself, and not describe a cellpair), From the above listed Dudeney patterns, show that most patterns can be described by only a single permutation in either direction, the patterns VII / X every column need a seperate permutation to describe the involved pairs, although noticed the central vertical symmetry which might be utilized (Left for some future date) 

self inverse permutation factor: 10 
Permutation which is it's own inverse ie: P[P[i]] = i  
There are 3 possibilities for pairing off the first element for a swap, with the other pair either in normal or reversed order when the first element is not involved there are (^{3}_{2}) = 3 posibilities for swapping pairs aside from the neutral. (see ptuple patterns article for generalisation of this argument) 

(0,1,2,3) (0,1,3,2) , (0,2,1,3) , (0,3,2,1) (1,0,2,3) , (2,1,0,3) , (3,1,2,0) (1,0,3,2) , (2,3,0,1) , (3,2,1,0) 

remarks  
Notice that the dudeney patterns all use merely the symmetric permutations in the above shown horizontal direction, ie the permutations (0,1,2,3) , (0,2,1,3) , (3,1,2,0) , (1,0,3,2) , (3,2,1,0) and (2,3,0,1) in fact only patterns XI uses tha asymmetric (0,3,2,1) and pattern XII the asymmetric (0,1,3,2) for some (yet unclear) reason the permutations (0,2,1,3), (1,0,2,3), (2,1,0,3) and (3,1,2,0) seem to be missing from the dudeney patterns, so the used permutations are: (0,1,2,3) , (0,1,3,2) , (0,3,2,1) , (1,0,3,2) , (3,2,1,0) and (2,3,0,1) note that (1,0,2,3) and (2,1,0,3) are mirrors of (0,1,3,2) and (0,3,2,1) respectively, and result in just horizontal mirriors of dudeney patterns XI and XII when used. while (0,2,1,3) is outer nutral and (3,1,2,0) is inner nutral, suggesting it is neccessairy to do bothe (ie (3,2,1,0)) or swap at least 1 inner to an outer position 
The Magic Encyclopedia Dudeney Patterns (note: generalisation of order 4 Dudeney Patterns in higer dimension) 


description  remark 
DP{[0,1,2,3],[0,1,2,3]}  Nutral mapping of cells onto themselfs 
Patterns involving a single 1agonal (note: IV (transposed) is isued as a sample (the other patterns can simularly be transposed) 

DP{[1,0,3,2],[0,1,2,3]} 
pairing of {{0,y],[0,y]} and {{2,y],[3,y]} Dudeney Pattern IV (transposed) 
DP{[0,1,2,3],[1,0,3,2]} 
pairing of {{x,0],[x,1]} and {{x,2],[x,3]} Dudeney Pattern IV 
DP{[0,1,2,3],[2,3,0,1]} 
pairing of {{x,0],[x,2]} and {{x,1],[x,3]} Dudeney Pattern V 
DP{[0,1,2,3],[3,2,1,0]} 
pairing of {{x,0],[x,3]} and {{x,1],[x,2]} Dudeney Pattern V1 
Patterns involving 2 1agonals  
DP{[1,0,3,2],[1,0,3,2]} 
pairing of {{0,0],[1,1]} and {{2,2],[3,2]} and mixed Dudeney Pattern II 
DP{[2,3,0,1],[2,3,0,1]}  Dudeney Pattern I 
DP{[3,2,0,1],[3,2,0,1]}  Dudeney Pattern III 
DP{[1,0,3,2],[0,3,2,1]}  Dudeney Pattern X1 
DP{[2,3,0,1],[0,1,3,2]}  Dudeney Pattern X1I 
adding dimension is simply done by inserting a selfinverse permutation in a "slot" opened by that dimension. the added dimension is as usual opening a "slot" on the right movein the permutation to another "slot" gives merely a transpositional variant 

DP{[1,0,3,2],[1,0,3,2],[0,1,2,3]}  Dudeney Pattern II in every xy plane 
DP{[1,0,3,2],[1,0,3,2],[1,0,3,2]}  Dudeney Pattern II in every xy yz and xz planes 
DP{[1,0,3,2],[1,0,3,2],[2,3,0,1]} 
Dudeney Pattern II in every xy plane Dudeney Pattern I in every xz and yz plane 
DP{[1,0,3,2],[1,0,3,2],[3,2,1,0]} 
Dudeney Pattern II in every xy plane Dudeney Pattern III in every xz and yz plane 
With the above notation regular camplementairy pair patterns can be handled however aside from seeing the square dudeny pattern when the (hyper)cube is viewed perpendicular to an orthogonal plane the above already show samples where in one direction there is no pairing at all (pe PD{[0,1,2,3],[0,1,2,3],[1,0,3,2]} which shows the dudeney IV pattern entirily in the z direction in either the xz and yz planes, while the xy planes show no paring at all. Reflecting upon this any combination of the 10 possible selfinverse permutations can be used in either x or y direction giving 10^{2} posibilities combined with either [1,0,3,2], [2,3,0,1] or [3,2,1,0] in the zdirection. More complex combination in the zdirection are possible given a permutation pair on the xy plane.  
Functional pair parametrisation Dispite the complex pair patterns which can be handled with the above method, already in the order 4 cube quite regular patterns can be found which defy this notation. When pe the position [i,j,k] paired of with position [3k,3j,3i] defies the notation. This twisted Dudeney VI pattern might be denotable by DP{[3,2,1,0]^{[2,1,0]}} as well as DP{[3z,3y,3x]} (probably however the latter is less general) NOTE: all this is preliminairy pending future investigation, (feel free to contribute) 
Dudeney Pattern Isomorphisms  

patterns  isomorphism 
group II ==> group I DP{[1,0,3,2],[1,0,3,2]} ==> DP{[2,3,0,1],[2,3,0,1]} 
IM{[0,2,1,3],[2,3,0,1]} [0,0] [2,3] [1,3] [3,0] [0,1] [2,2] [1,1] [3,1] [0,2] [2,1] [1,1] [3,2] [0,3] [2,0] [1,0] [3,3] 
group II ==> group III DP{[1,0,3,2],[1,0,3,2]} ==> DP{[3,2,1,0],[3,2,1,0]} 
IM{[0,3,2,1],[2,3,0,1]} [0,0] [3,2] [2,0] [1,2] [0,1] [3,3] [2,1] [1,3] [0,2] [3,0] [2,2] [1,0] [0,3] [3,1] [2,3] [1,1] 
group IV ==> group V DP{[0,1,2,3],[1,0,3,2]} ==> DP{[0,1,2,3],[2,3,0,1]} 
IM{[0,1,2,3],[0,2,1,3]} [0,0] [1,0] [2,0] [3,0] [0,2] [1,2] [2,2] [3,2] [0,1] [1,1] [2,1] [3,1] [0,3] [1,3] [2,3] [3,3] 
group IV ==> group VI DP{[0,1,2,3],[1,0,3,2]} ==> DP{[0,1,2,3],[3,2,1,0]} 
IM{[0,1,2,3],[0,3,2,1]} [0,0] [1,0] [2,0] [3,0] [0,3] [1,3] [2,3] [3,3] [0,2] [1,2] [2,2] [3,2] [0,1] [1,1] [2,1] [3,1] 
group VII ==> group VIII DP{[0,1,2,3],{(3,2,1,0),(1,0,3,2),(1,0,3,2),(3,2,1,0)}} ==> DP{[0,1,2,3],{(2,3,1,0),(3,2,1,0),(3,2,1,0),(2,3,1,0)}} 
IM{[0,1,2,3],{[0,1,3,2],[0,3,2,1],[0,3,2,1],[0,1,3,2]} [0,0] [1,0] [2,0] [3,0] [0,1] [1,3] [2,3] [3,1] [0,3] [1,2] [2,2] [3,3] [0,2] [1,1] [2,1] [3,2] 
group VII ==> group IX DP{[0,1,2,3],{(3,2,1,0),(1,0,3,2),(1,0,3,2),(3,2,1,0)}} ==> DP{[0,1,2,3],{(1,0,3,2),(3,2,1,0),(3,2,1,0),(1,0,3,2)}} 
IM{[0,1,2,3],[0,3,2,1]} [0,0] [1,0] [2,0] [3,0] [0,3] [1,3] [2,3] [3,3] [0,2] [1,2] [2,2] [3,2] [0,1] [1,1] [2,1] [3,1] 
group VII ==> group X DP{[0,1,2,3],{(3,2,1,0),(1,0,3,2),(1,0,3,2),(3,2,1,0)}} ==> DP{[0,1,2,3],{(3,2,1,0),(2,3,0,1),(2,3,0,1),(3,2,1,0)}} 
IM{[0,1,2,3],{[0,1,2,3],[0,2,1,3],[0,2,1,3],[0,1,2,3]} [0,0] [1,0] [2,0] [3,0] [0,1] [1,2] [2,2] [3,1] [0,2] [1,1] [2,1] [3,2] [0,3] [1,3] [2,3] [3,3] 
group XI ==> group XII DP{[1,0,3,2],[0,3,2,1]} ==> DP{[2,3,0,1],[0,1,3,2]} 
IM{[0,3,2,1],[0,2,3,1]} [0,0] [2,0] [1,0] [3,0] [0,1] [2,3] [1,3] [3,1] [0,2] [2,1] [1,1] [3,2] [0,3] [2,2] [1,2] [3,3] 
group II ==> group IV DP{[1,0,3,2],[1,0,3,2]} ==> DP{[0,1,2,3],[1,0,2,3]} 
[0,0] [1,0] [2,0] [3,0] [1,1] [0,1] [3,1] [2,1] [1,2] [0,2] [3,2] [2,2] [0,3] [1,3] [2,3] [3,3] 
group II ==> group VII DP{[1,0,3,2],[1,0,3,2]} ==> DP{[0,1,2,3],{(3,2,1,0),(1,0,3,2),(1,0,3,2),(3,2,1,0)}} 
[0,0] [1,0] [2,0] [3,0] [1,3] [0,1] [3,1] [2,3] [0,2] [1,2] [2,2] [3,2] [1,1] [0,3] [3,3] [2,1] 
group II ==> group XI DP{[1,0,3,2],[1,0,3,2]} ==> DP{[1,0,3,2],[0,3,2,1]} 
[0,0] [1,1] [2,1] [3,0] [0,1] [1,2] [2,2] [3,1] [0,2] [1,2] [2,2] [3,2] [0,3] [1,0] [2,0] [3,3] 