  We analyze the set of real and complex Hadamard matrices with additional
symmetry constrains. In particular, we link the problem of existence of
maximally entangled multipartite states of $2k$ subsystems with $d$ levels each
to the set of complex Hadamard matrices of order $N=d^k$. To this end, we
investigate possible subsets of such matrices which are, dual, strongly dual
($H=H^{\rm R}$ or $H=H^{\rm\Gamma}$), two-unitary ($H^R$ and $H^{\Gamma}$ are
unitary), or $k$-unitary. Here $X^{\rm R}$ denotes reshuffling of a matrix $X$
describing a bipartite system, and $X^{\rm \Gamma}$ its partial transpose. Such
matrices find several applications in quantum many-body theory, tensor networks
and classification of multipartite quantum entanglement and imply a broad class
of analytically solvable quantum models in $1+1$ dimensions.
