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We have performed high-fidelity simulations of turbulent open-channel flows over submerged rigid canopies made of cylindrical filaments of fixed length $l=0.25H$ ($H$ being the domain depth) mounted on the wall with an angle of inclination $θ$. The inclination is the free parameter that sets the density of the canopy by varying its frontal area. The density of the canopy, based on the solidity parameter $λ$, is a widely accepted criterion defining the ongoing canopy flow regime, with low values ($λ\ll 0.1$) indicating the sparse regime, and higher values ($λ> 0.1$) the dense regime. All the numerical predictions have been obtained considering the same nominal bulk Reynolds number (i.e. $Re_b=U_b H/ν= 6000$). We consider nine configurations of canopies, with $θ$ varying symmetrically around $0°$ in the range $θ\in [\pm 78.5°]$, where positive angles define canopies inclined in the flow direction (with the grain) and $θ=0°$ corresponds to the wall-normally mounted canopy. The study compares canopies with identical solidity obtained inclining the filaments in opposite angles and assesses the efficacy of the solidity as a representative parameter. It is found that when the canopy is inclined, the actual flow regime differs substantially from the one of a straight canopy that shares the same solidity indicating that criteria solely based on this parameter are not robust. Finally, a new phenomenological model describing the interaction between the coherent structures populating the canopy region and the outer flow is given.