This paper presents a closed-form expression for the Jones matrix of a dual-polarized inverted-vee dipole antenna based on the Lorentz reciprocity theorem and the basic rules of electromagnetic refraction. The expression is used to determine the intrinsic cross-polarization ratio (IXR) as a function of droop angle, position of the source in the sky, antenna height, frequency, and reflection coefficient of the underlying ground. The expression is verified using full-wave simulations with a method-of-moments solver, showing very good agreement. It explains the increase in the IXR when the antenna is placed over a perfect electric ground plane. This result is used to explain the polarization properties of the Square Kilometre Array Log-periodic Antenna. Through the LOw-Frequency ARray Low-Band Antenna (LOFAR-LBA), the importance of the size of the ground plane is explained. Finally, design consideration for high polarization purity antennas is discussed.