For a bounded domain Ω in R^N with Lipschitz boundary Γ = ∂Ω and a relatively open and non-empty subset Γ_t of Γ, we prove the existence of a positive constant c such that inequality c||T ||_L²(Ω,R^N×N) ≤ ||sym T ||_L²(Ω,R^N×N) + || Curl T ||_L²(Ω,R^{N×N(N−1)/2}) holds for all tensor fields T ∈ H(Curl; Γ_t,Ω,R^N×N), this is, for all T : Ω → R^N×N which are square-integrable and possess a row-wise square-integrable rotation tensor field Curl T : Ω → R^{N×N(N−1)/2} and vanishing row-wise tangential trace on Γ_t. For compatible tensor fields T = ∇v with v ∈ H¹(Ω,R^N) having vanishing tangential Neumann trace on Γ_t the inequality reduces to a non-standard variant of the first Korn inequality since Curl T = 0, while for skew-symmetric tensor fields T the Poincaré inequality is recovered. If Γ_t = ∅, our estimate still holds at least for simply connected Ω and for all tensor fields T ∈ H(Curl; Ω,R^N×N) which are L2(Ω,R^N×N)-perpendicular to so(N), i.e., to all skew-symmetric constant tensors.