A detailed analysis of a newly appeared nonconforming rotated Q1 finite element is presented. It is proved that this element is convergent on quadrilateral meshes for the general second order elliptic problem provided a condition on the mesh subdivision is imposed. Some optimal quadrature schemes to calculate the stiffness matrix are proposed, including a two-point scheme that excludes even a P1 unisolvent set. Finally, the rotated Q1 element is used to approximate the Reissner-Mindlin plate. A new locking free lowest order rectangular element is obtained.