This is just my view and maybe not correct mathematically or puzzlogically.

If an axis is a set of points invariant by the rotation, an axis of rotation of 4D cube forms a plane(a 2D entity).

Ignoring cw or ccw, there are 3 90Â° rotations about axes perpendicular to faces of a cube

**Code:**

f(x,y,z) = (-y, x, z) # about z

f(x,y,z) = ( x,-z, y) # about x

f(x,y,z) = ( z, y,-x) # about y

Ignoring cw or ccw, there are 6 90Â° rotations about axes

perpendicular to cells of a tesseract **Code:**

f(w,x,y,z) = (-x ,w, y, z) # about yz-plane

f(w,x,y,z) = ( w,-y, x, z) # about zw-plane

f(w,x,y,z) = ( w, x,-z, y) # about wx-plane

f(w,x,y,z) = ( z, x, y,-w) # about xy-plane

f(w,x,y,z) = ( w,-z, y, x) # about wy-plane

f(w,x,y,z) = ( y, x,-w, z) # about xz-plane

On my 8.1.1(2x2x2x2), clicking on two side of a face(entity connecting 2 cells) are same rotations but work on different layers.

So doing them to all layers result to the same orientation of the whole tesseract.

for example:

**Code:**

AG',//axis : wx , layer: z>=0

HB',//axis : wx , layer: z< 0

GA, //axis : wx , layer: y>=0

BH, //axis : wx , layer: y< 0

Another comparison in terms of group :

Orientations of a cube form a group with 24 elements divided into 5 conjugacy classes.

**Code:**

--+-------------------+-----

1 | Identity | 1

2 | 90Â° face turns | 6

3 | 180Â° face turns | 3

4 | 120Â° vertex turns | 8

5 | 180Â° edge turns | 6

--+-------------------+-----

| total | 24

--+-------------------+-----

Orientations of a tesseract form a group with 192 elements divided into 13 conjugacy classes.

**Code:**

--+-------------------+-----

1| Identity : 1

2| Center : 1

3| modulo 2 : 6

4| modulo 4 : 6

5| modulo 4 : 6

6| modulo 2 : 12

7| modulo 4 : 12

8| modulo 4 : 12

9| modulo 2 : 24

10| modulo 8 : 24

11| modulo 8 : 24

12| modulo 3 : 32

13| modulo 6 : 32

--+-------------------+-----

| total |192

--+-------------------+-----

They include classes corresponding to face,vertex and edge turns of cube but others too.

Which interests me particularly is modulo 6 and modulo 8 operations. Theoretically it's possible to make puzzles with only these operations(maybe they already exist?) But I don't know how to describe them in words or in analogy with 3D. For example this operation has a modulo8-cycle.

**Code:**

f(w,x,y,z) = (-z,w,x,y)

What remains invariant ? Or how can I describe the equation of the axis ?

I have no idea.