A 4-node hybrid stress quadrilateral element is presented with compatible isoparametric bilinear displacements and a new version of 5-parameter stresses which satisfies both self-equilibrium equations and an energy orthogonal relation between stress terms and Wilson incompatible strains. Based on different element formulations, another two new elements are also derived and shown to be identical to the first element. Numerical benchmark experiments show that the three equivalent elements possess high accuracy at coarse meshes, are frame invariant and free from Poisson locking at the nearly incompressible limit. Since the 5-parameter stress mode used does not have uncoupled constant terms, the elements do not pass the patch test.