This paper extends our earlier approach [cf. A. Thyaharaja, Phys. Plasmas 17, 032503 (2010) and Krishnaswami et al., Phys. Plasmas 23, 022308 (2016)] to obtaining à priori bounds on enstrophy in neutral fluids and ideal magnetohydrodynamics. This results in a far-reaching local, three-dimensional, non-linear, dispersive generalization of a KdV-type regularization to compressible/incompressible dissipationless 2-fluid plasmas and models derived therefrom (quasi-neutral, Hall, and ideal MHD). It involves the introduction of vortical and magnetic "twirl" terms λ l 2 (w l + (q l / m l) B) × (▿ × w l) in the ion/electron velocity equations (l = i, e) where w l are vorticities. The cut-off lengths λl and number densities nl must satisfy λ l 2 n l = C l, where Cl are constants. A novel feature is that the "flow" current σ l q l n l v l in Ampère's law is augmented by a solenoidal "twirl" current σ l ▿ × ▿ × λ l 2 j flow, l. The resulting equations imply conserved linear and angular momenta and a positive definite swirl energy density E∗ which includes an enstrophic contribution σ l (1 / 2) λ l 2 ρ l w l 2. It is shown that the equations admit a Hamiltonian-Poisson bracket formulation. Furthermore, singularities in ▿ × B are conservatively regularized by adding (λ B 2 / 2 μ 0) (▿ × B) 2 to E∗. Finally, it is proved that among regularizations that admit a Hamiltonian formulation and preserve the continuity equations along with the symmetries of the ideal model, the twirl term is unique and minimal in non-linearity and space derivatives of velocities.