Two affine Hadamard families $H_1(R_1)$ and $H_2(R_2)$, stemming from $N\times N$ complex Hadamard matrices $H_1$ and $H_2$, associated with real matrix spaces $R_1$ and $R_2$ of the same dimension, are called cognate if $$\forall\,B\in H_2(R_2)\,\exists\,A\in H_1(R_1):B\simeq A^{\rm T},$$ $$\forall\,A\in H_1(R_1)\,\exists\,B\in H_2(R_2):A\simeq B^{\rm T}.$$ The family $H(R)$ is called self-cognate if $$\forall\,B\in H(R)\,\exists\,A\in H(R):B\simeq A^{\rm T}.$$