Konrad wrote:

Here is a double swap of 4x4x4 edges (= seemingly a swap of two 3x3x3 edges) that does not change the orientation of centres on a Super Cube:

Gelatinbrain notation:

B'&2, R, U', B, R', U, B&2, F&2, U', R, B', U, R', F'&2

Many thanks Konrad. This algorithm was exactly what I was needing (and it does not have the side effect of the r2 U2 r2 u2 r2 u2 U2 algorithm that I was using).

This 4x4x4 fisher cube convinced me that 4x4x4 supercube is probably a great puzzle. I will buy the sticker to build one.