The Haagerup-set [62] of a complex Hadamard matrix $H$ of order $N$ is the set $$\Lambda(H) = \Big\{h_{ij}h_{kl}\bar{h}_{il}\bar{h}_{kj} : i,j,k,l\in\{1,...,N\}\Big\}$$ where $h_{..}$ is appropriate matrix element and bar over $h$ denotes complex conjugate.
Haagerup-set is invariant under the equivalence relation, that is, two matrices with different $\Lambda$'s are inequivalent.
For instance $F_2 \otimes F_2 \not\simeq F_4$ as $\Lambda(F_2) = \{\pm 1\} \subsetneq \{\pm 1, \pm i\}=\Lambda(F_4)$.
See also matrix fingerprint and rectangular rank profile.