Two affine Hadamard families stemming from dephased $N\times N$ complex Hadamard matrices $H_1$ and $H_2$ $:$ $H_1(R_1)$ and $H_2(R_2)$, associated with real matrix spaces $R_1$ and $R_2$ of the same dimension, are called permutation equivalent if there exist two permutation matrices $P_r$ and $P_c$ such that $$H_2(R_2)\simeq P_r\cdot H_1(R_1)\cdot P_c,$$ i.e. there is one-to-one correspondence, by row and column permutation, between the elements of $H_1(R_1)$ and $H_2(R_2)$.